The washing machine drain pipe backs up during the rinse cycle but not when I run the cold water hose down the pipe on high flow, how can that be?

Because the rinse cycle is under more pressure. There is a small blockage in the drain - look for the nearest cleanout, hopefully below in a basement?

Your wash machine is probably putting out a higher pressure than the garden hose. Or you may have a blocked drain pipe the only thing will solve the problem to snake and clear the drain line.

If the line is cleared you have two options.

1. Seal off the washer hose with a compression fitting, see the image below.

2. Try inserting apiece of 3/4" copper pipe about 8" long into the discharge hose and fasten it with a hose clamp. This will restrict the flow and hopefully your drain will handle the flow, Good luck

John