Your opening paragraph does not feel like first person. It would work fine in third person, but not in first, and most definitely not in first person from a child's point of view.
What young child would mentally refer to his own hometown as "the city of rainbow dreams"? What young child would use language like "A gentle snowfall covered the bare trees and smothered the few, dainty flowers that were still alive despite the chilly weather"? How does young Eusine know in detail what those older children are going to do when they go inside? That paragraph is so blatantly third-person in both diction and subject that it feels extremely disorienting when it suddenly starts speaking of "my house". In fact, the story can be split into the third-person first paragraph and the first-person remainder, with the writing styles of these two sections having absolutely nothing in common. The first paragraph is flowery, omniscient, mature, descriptive; the rest is simplistic, limited, naïve, barebones, save for a couple of jarringly formal sentences in between. Pick one style and stick with it - and since you're trying to write first person from a child's point of view, the style of the first paragraph simply cannot work. Though to be honest, actually I agree with #1 ***** fan: you really don't seem all that good at first person. The style of the first paragraph flows much better and feels more refined than the rest, and I can't help thinking it's because that's a style that comes much more naturally to you. Perhaps it would work better with Eusine as an adult, when his inner monologue could be forgiven for being somewhat flowery since he seems like that kind of guy, though. :/
“So, you’re telling me that you’re going to search for that legend of a Pokemon again!? And you don’t know when you’re coming back!” I could hear mommy shout at daddy from another room.
“I told you I’ll be back in a month or two, woman!” daddy raised his voice for the first time in their argument.
There was a petrifying silence and no one spoke.
This seems like a
mighty odd place for a dramatic silence. Dramatic silences are caused when one person is in too much of a shock to reply immediately and as such should follow dramatic remarks, not a simple declaration of when he will be back which, judging from the "I told you", she already knows anyway. In this context, it sounds more like an "I can't think of anything to say" silence, which is not very petrifying.
A child is not likely to internally describe anything as "a petrifying silence"; again, something that would work great and provide nice imagery in third person, but not in the first-person POV of a child. "No one spoke" is similarly rather too formally worded for a child who still refers to his parents as "Mommy" and "Daddy" (when these are used as names, by the way, as you're doing, they should be capitalized). "There was a petrifying silence and no one spoke" is also redundant; if there was a petrifying silence, it follows that nobody is speaking.
She stood resolute with a look on her face that could scare a Ghastly.
"Resolute" is also a word a child is very unlikely to use internally.
And the Pokémon is called Gastly, not Ghastly. :/ Really, the least you could do is make sure you're spelling Pokémon names right when writing a Pokémon fanfic.
The argument feels rather wooden; I think it's because there is no actual character emotion in it, just one soap opera argument cliché after another. "You're going to leave me behind!" "He's your son, Arthur!" "I'm not making a big deal out of nothing!" "You would never understand!" "Don't try to hug me now!" "Don't be surprised when I lock the doors on you!" The actual personal issues at stake are drowned underneath all these clichés - Arthur feels like Generic Soap Opera Father, and Eusine's mother feels like Generic Soap Opera Mother, and they're having Generic Soap Opera Argument. It makes it impossible to relate to the characters as characters, properly understand what drives them and actually feel the tension between them. Why does Arthur search for Suicune? Why is it so extremely important to him? To what lengths has he gone with it before? Why does this particular time tip his wife over the edge? What does she worry about beyond just his leaving? How does she actually feel about him? If the argument were written better, in a less clichéd manner, and you had more indicators of details like this, it would be much easier to take it seriously.
Again, there was the calm before the storm, where no one spoke. Then, there was the storm -- the shouting, the arguing, and the crying.
Yet again, what child thinks in words like this?
Aside from that, using "there was" is a very weak wording, particularly when it's the verb in two sentences in a row. It makes them seem bland and routine - "first there was this, then there was this" - instead of the emotional effect you should be going for here.
And really, this metaphor isn't even appropriate here. The calm before the storm refers to calm, well,
before the storm, not in the middle of it. :/
I couldn’t take it so I shut my door and hid under the cozy covers of my bed.
This is the sentence where I concluded you don't seem to be very good at first person. Everything that preceded it is perfectly objective and emotionless - and now you say Eusine "couldn't take it"? How on earth can we have been in his head during the whole argument and not noticed even a hint that he's scared or worried? I assumed, thanks to the lack of emotion in him during the actual argument, that he was really detached, so used to seeing them fight that he could just shrug it off, or else somehow had absolutely no idea what was going on or that it was anything serious. If it is really distressing for him to watch the argument,
show it while that's happening - have him notice with fear that his daddy has a scary look on his face, have him curl up behind the bedroom door again while clutching it tightly with his fingers, have him close his eyes or describe how he flees to his room to try to block out the sound. He shut the door? Shouldn't a frightened child be a little less careful than that - slamming it, perhaps? His bed covers are cozy? How can he be thinking about coziness when he's in severe distress? He should be cowering, pulling the covers over his head, squeezing his eyes shut in the darkness while his own breathing almost drowns out his parents' voices, waiting for it to be over.
And that's another thing - that really is what he ought to be doing. Why does he just pick up a book and start reading it? The moment after he's back in his room and under the covers, supposedly "hiding", he completely stops seeming upset again. He just starts reading a book, noticing the pretty wolf (why does everyone think Suicune looks anything like a wolf, anyway?), trying to read its name, smiling, yawning, going to sleep as if nothing were more natural. He should be too worked up to do any of that so casually - at most I could picture him flipping restlessly through the book to divert his attention.
As far as we can tell, Eusine is completely emotionless, or at least has the emotional memory of the proverbial goldfish. That's not a good thing for a prologue like this, which seems to be trying to rely on emotion to provide the intended effect. Show us more emotion and this would be a whole lot more interesting.
To be honest, I think this prologue really would probably work better in third person, since first-person young child POVs are really hard to do well even if you did try to stick with it - it's so easy for it to sound either cheesy and fake or way too mature. But if the rest of the story is first-person from adult Eusine's point of view, I suppose you'd want to stick with that for the prologue too. :/ Regardless, the writing style here really doesn't work. In the first paragraph the language is too mature and flowery and the point of view is too omniscient; in the rest of it there is too little interesting detail and emotion to make the story gripping and involving.
Others have commented on your grammar and punctuation use, but it too could use some refinement.
). 
This is all personal opinion, so take it whatever way you want.
