You would win your bet. I've seen Autozone stadium from the outside but didn't notice any parking anywhere near it. I've been to the Vet in Philly (sort of suburban, in South Philly), but I think they have a new stadium now. Not sure where it is. I've been to Candlestick, which is darn near rural ... but with a huge parking lot. I think it's being replaced too. Been to Turner Field, downtown, but not the new one. I've been to Cardinals' Stadium in St. Louis (downtown) but had to park a long way away. Been to Dodgers' Stadium. It's sorta downtown LA, but also sort of in a remote location. Easy parking. That's all I can remember at the moment.
You win the bet because I haven't been to a new downtown stadium, either minor or major leagues. Regarding Hoover ... it depends on where you're coming from. It's a few miles further for me since I'm coming from the north, but should be about the same as downtown if you're coming from most of Mississippi. If you're coming from the south, Hoover is a easier to get to than downtown.
I'm not sure exactly where the new B'ham stadium. If it's close enough, you could use the BJCC parking area, if they don't have another event going on. During normal times though, traffic shouldn't be a huge problem simply because not all that many people go to minor league games ... probably typically 3-4,000. The tournament would be a different matter though.