Most nearby, off-site hotels offer some sort of transportation. Personally, I've never heard of any off-site hotel that provides transportation to any Disney Resort. I'd guess that Disney wouldn't allow that to happen.
The key to off-site transportation is knowing the details. Most will only make 2 or 3 trips each way each day. Often, they only do to one destination at Disney World. Typically, they will go to Epcot. From there, you must transfer to Disney Transportation to get anywhere else. Also, some may make stops at other hotels along the way. Finally, sometimes you are required to sign-up in advance if you want a ride.
In between Disney Resorts and completely off-site hotels sits the Disney Springs Hotels. These seven hotels are located on Disney property adjacent to the Marketplace section of Disney Springs. They are not owned nor operated by Disney. Although prices vary, they are typically less expensive than comparable Disney Resorts.
Disney Springs Hotels Marketing website:
http://disneyspringshotels.com/
The Disney Springs Hotels provide bus transportation every 30 minutes to all four theme parks. They also provide buses to Disney Springs in the evening...although they are all within a 15 minute walk. Each hotel is on two routes. Typically, one will go to Epcot followed by Disney's Hollywood Studios. The other will run to the Transportation and Ticket Center (transfer to boat/monorail for Magic Kingdom) followed by Disney's Animal Kingdom.
Each route does stop at 3 or 4 hotels prior to heading to the theme parks. As the hotels are next to each other, this procedure does not take long. (The exception would be if a wheelchair or
ECV must be loaded.) As the hotels are smaller than most Disney Resorts, there usually are only a few people who get off/on at each stop.
The Disney Springs Hotels might work for your situation. Your parents would be able to hop on a bus and get off at the first stop (the Transportation and Ticket Center). Then, they can take a scenic monorail trip over to Disney's Contemporary Resort to Bay Lake Tower.