NTA if you don’t apologize but it will confuse and stress him more.
My guess is he was trying at banter and the fact that people only snickered at yours made him think yours went over some line that his didn’t.
I’m autistic too, and what Parker needs if anything is a public apology, not a private one like he’s asking for.
And I don’t mean you owe him a public apology. I’m saying the scope of his perceived loss of status (the embarassment) is the group, not your one on one relationship.
What Parker needs to learn is that group karma needs to be processed in the group. What he needs most of all is to say some shit to you, and for the group to snicker in response. An excellent second best would be for you to publicly compliment him.
I’m autistic but I’m also 40 so I’ve had a lot of time to glean the rules.
Next time you see him with shots of that Spider Man guy, maybe compliment the shots. That’s your best move for creating peace with Parker. And hope to god that he’s not too dense to accept it gracefully. He probably is.