What type of company should I call to arrange moving an elderly couple and hospital beds from the second floor to the first floor?

A friend had this issue in her home, and called a local moving service to move the bed from upstairs down to the first floor. (I assume you're referring to moving the actual furniture, as opposed to physically moving the parents?

I can see that this could be a tough problem requiring a bit of coordination.
If it’s both parents and beds I think you would need two separate services.
In addition to the moving company Garden suggested you might also try checking with a medical supply that delivers and services hospital beds to move the beds. To move your parents you might try the fire department or EMS or even the transport service you have used before. Be sure to come back and let us know how you you get this accomplished.

Thanks for your responses, GardenArtist and 97yroldmom, I thought this might be the case. For the beds I will use a moving company and for parents the transportation company. It will require very close coordination to get all this to happen at the same time.