It sounds like you've been doing too many things and that you started way too soon. Age 2.5 is way too young for most kids, especially after you figured out he couldn't manage it.
I think it's a huge mistake to let him sit in his poop until he is sore and red! That's just not healthy.
Sitting on the potty for an hour is absolutely not productive. It's punishment, and you cannot win a battle where he has complete control. Same thing with making him sit in the park and watch the other kids. Watching other kids is not connected with potty training at all. I'm not sure you meant that you made him sit there because he pooped in his pants in the park - in which case you had him sitting in his own mess. Or, if you made him sit in the park because he pooped in his pants earlier in the day, then you are separating the punishment from the act by too long a period of time. Either way, not productive.
Some kids just take longer than others. Some kids really have issues parting with their poop. I'd suggest you just put him back in diapers or pull-ups, and stop the arguing. It's a battle of wills right now, and he's winning.
I think it's fine that you make him clean up his own poop in the pull-up, as long as you supervise. Be matter-of-fact about it. Don't make it fun, and don't make it a lecture. The less attention he gets for this, the better.
The summer is a long way away in the life of a child. I wouldn't focus on the pool right now, and I wouldn't discuss it. He knows what you want, and he knows it irritates you when he does something different. He's getting more attention from you when he poops in his pants and when he laughs at you than he does with prizes and sticker charts. I'd put it all away and just let him know that whenever he is ready, he can tell you.
Let things cool off for a while. Stop talking about it. He may just do it by himself. Maybe not. Then you can SLOWLY introduce the idea that cleaning him up takes time away from things HE'D like to do. I'd put him in the shower and let him wash himself - you can read a book nearby to be sure he's safe, but don't engage him in conversation. Hygiene needs to become a required activity that is within his control in terms of how long it takes. When he completes his basic tasks (peeing, pooping, brushing teeth, cleaning himself), then there's time for playing and recreation.
Same goes for other chores - clear the table after dinner and get the dishes done, THEN there's time for a story and some special time. If he doesn't help in some small way with the clearing of dishes, then he has CHOSEN to give up story time. If he doesn't pick up his toys, then he has CHOSEN not to have TV time. If he continues to poop in his pants, then you're not able to do other things with him. When he's older, it will be different. You have to make it more matter-of-fact and less judgmental/punitive. Because it's turning into a control session - and they say that kids control what they eat, when they sleep, and when they pee/poop.
Good luck.
ETA - as soon as I posted this, the other responses popped up. I have to say I disagree with humiliating him in front of his friends. Shame is not a positive emotion or motivator - it makes it okay to bully and humiliate someone. And I don't believe in spanking - again, it just teaches that it's okay for a big person to hit a smaller person. Not the message you want to give him. Just ignore his laughing at you - leave the room. Do not reward him for it.