Can I put the Barka Parka on a frame?

NO. Never!
A Barka Parka is designed as a stand alone bed which provides support and comfort.

The frame will work directly against this and besides the Barka Parka will tend to fall off the sides of the frame and the beans will flow to the lowest point meaning that all your dog will get out of their Barka Parka is three thicknesses of fabric.