Will I lose everything if I put a new Apple ID on my phone?

So I've been using my parents' Apple ID for years now and I'd like my own. All my music, apps, etc. Are connected to my parents' account. To be honest I'm not too worried about my music; I mostly use Spotify anyway & I can just buy new stuff.
What I'm worried about (this is pathetic lol) is my Sims Freeplay game. I've been playing this game for ages now and to lose my progress would honestly be kind of devastating.
However, I really want my own Apple ID. If I were to make a new one and use that on my iPhone, would I lose my progress on the Sims Freeplay game? If I connected it to a Game Center (I currently don't have the Game Center) would it delete everything or would it save everything?

Yes

Your progress shouldn't be erased. However, you typically can't transfer save progress between Apple ID accounts.
So in the future, if you get a new phone or have to factory reset your phone, you will loose all data.

AFAIK, simply signing out will not erase your data.

If you've made purchases through iTunes or App Store using your parents credit card, all you need to do is set up Family Sharing between your new Apple ID and theirs. Just go to Settings, then select their Apple ID and go to Family Sharing and set it up there. That way you can have your own Apple ID but still be connected to their payment info, so that you can get new songs they purchased, Apps they purchased, etc.

As far as the Sims FreePlay thing goes, it's been a long time since I played that game but, as far as I'm aware, it's saved through Facebook, not through a specific Apple ID. If you haven't linked your Facebook account to it, go into the game and do so then you should be able to open your progress on a new Apple ID or device. Like I said though, it's been a long time and things might've changed but I'm sure they haven't changed that.

I don't know the specifics about how that game works in terms of what accounts it connects to, but the data for it should be on the phone regardless of Apple ID that is signed in.

If you switch to a new Apple ID, you lose everything that was purchased with the previous Apple ID.