There's no hard and fast rule on what time of day the parking booths are unattended, AFAIK. Last night at about 7:30 they were unattended when we drove into the TTC, 2.5 hours before closing. In the past we've seen the booths attended as late as an hour before closing and as early as about four hours prior.
We're also British, and Passholders. If annual passes go on sale again before your visit, it may be worth you working out if it would be worth it. If you're visiting for two weeks it may well be. Also look at the typical fares for Lyft/Uber from your accommodation vs car rentals/Turo + parking.
As posters above have said, Disney Springs free parking is possible for the parks, but you need to add an hour onto your journey to the parks, and another hour for your journey back. It may be less, but not by much.
You could also consider making dining reservations at parkside resorts on some days. If you planned on eating at any of the MK or Epcot resort hotels you *might* find it convenient to time these to coincide with your park visits. Parking is free at hotels with a dining reservation, and security will generally wave you into the parking lot as early as 2-3 hours before your reservation. From Wilnerness, Contemporary, Grand Flo and Poly it's a short walk, monorail or boat ride to MK, and from Boardwalk, Yacht & Beach, Swan & Dolphin, Riviera and Caribbean you have super easy access to both Epcot and DHS.
