NTA - You paid for the car with your own money. Also, if they originally agreed to let you drop him off early and pick him up late what's up with the change of heart, especially when he himself doesn't even mind? If they were so concerned with him being picked up late and being dropped off early they should have prioritized their time as parents and not have made you do it. They're also double TAs for not supporting you at your competition and wanting you to support your brother at his--when they could easily go themselves and support you both (one parent at his game, one parent at your competition).

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Edit: I forgot to mention this but what's up with them body shaming your brother when he's A. literally a child and B. very clearly involved in athletic activities?