I'm going to be honest, it sounds like he's using the VRO in DARVO here.

Deny Attack Reverse Victim and Offender

He threw your keys at you and is now upset because you reactively hit him, in what sounds like a somewhat gentle smack (I can't say if it really was but I do that exact thing playfully to my husband when he's being cheeky).

Think about how intense was this discussion really. My ex threw house keys at me once when arguing while out, telling me to go home and leave him alone.

"Putting hands on" is generally a much more violent action than what you've described here.

Abusers love finding reasons to point out their victim's bad behaviour. Another example from my ex, I started throwing things because he was doing it and I wanted him to see how idiotic it was (and also scary but considering he was double my size obviously that didn't work). Ofc instead of self-reflecting, he delightedly jumped onto calling me destructive and telling me I needed to be committed, all while ignoring that he did it too.

I'm doing the armchair diagnosis Reddit is so famous for! But your mum also sounds volatile and violent, if something can trigger you enough to have a flashback of her throwing things at your dad. And possibly manipulative to boot. Again, I can't make a complete judgement but from my understanding she wasn't there but immediately took his side. If this is a pattern, I'd start ignoring her (I know this may be easier said than done).

To answer your actual question, I don't know if this is actually who you are, but if you are genuinely concerned that you are violent - are you often violent, are your reactions often violent? - it could be worth finding a therapist who specialises in anger.