I think the ‘park approved’ is a bit of overkill on their part. There is no process for ‘park approval‘ so there are no official ‘park approved’ wheelchairs,
ECVs or other equipment. I think what the rental companies mean by it is that their devices are not bigger than the maximum size Disney allows.
Walkers are generally folded and held the same way strollers are on the bus. Some people attach the Walker to their
ECV or wheelchair (that may or may not work depending on how far it sticks out and how well the driver can fasten the tie down straps to secure the ECV with the Walker on back.
This appears to be the walker model Gold is using (or a very similar one)
Vive walker
The link has a picture of it folded, but it’s shown being put into a trunk, so maybe not the clearest picture. I’m sure Gold would give you further information if you contact them, but I’m quite sure you won’t have an issue with it on the bus or in the parks.