Mob, just have a look at some of the hotels around the airport - my parents have managed a few times to bag a deal where they book and pay for a room for the night before their flight and then get to park the car at the hotel for a fortnight as part of the deal.
That was Gatwick IIRC but presumably others do similar things?
If not, then how about an offsite park & ride? Again, I've only done this at Manchester but there must be alternatives in Birmingham. You have to leave your keys but it's only so they can move your car back to the front when you're due to land.
Take a Post-It note with you, write the mileage on it when you park and stick it over the speedo. If nobody moves the car, it won't matter.
Just how many Yorkshire sheep can you fit inside one exhaust?