Yes they all do fold. Go to the website for abbf and they show exactly which models they rent. Side by side, front to back and also a sit n stand. Then go to your local babies r us and check out each model to see which one you think best meets your needs. It depends on your children (esp ages), your touring style and your main means on transportation which stroller will best suit your needs. You may also get something on sale or at a consignment shop that costs less than either abbf rental or the park rental.
For us - once the youngest was 2 1/2 renting at the parks was the best move for us when staying at a resort with bus as primary mode of transportation. At an epcot or mk resort with monorial or boat as transportation a personal (or rental) stroller was great. My girls are 2 years apart so a sit and stand was wonderful for us.
TJ