Solved

Parts needed to repair cars increasing each race

How many parts are needed depends also on the track you just raced. Monaco is really easy on the car and thus requires very few parts to repair afterwards. Monza (Italy) is, again, the complete opposite.

As intermediate solution you could still use token from the sponsors to buy some parts. In the meantime build up the Manufacturing building ASAP if you haven't done so. Then keep the Design building 1-2 levels below your team level, this building influences the most how parts expensive your car repairs are.
Probably because of rounding especially in the level 12-15 you need a good deal more parts (20-30/season with my old team) than you can manufacture if you have both buildings maxed out. So either you make sure you can spend the token and thus make sure you budget your needs in between engines, cash and parts or sacrifice a few design points. Then there is indeed reliability but if this haven't got a boost I missed to notice it's not that much.

Edit: Another emergency help is not repairing for the last race(s). After the last one of the season you get a new car and thus you need no parts no matter how damaged the old one was. Of course it means sacrificing speed there.