If you move the parents and eggs they "might" stay with the eggs but then again they might freak out and not. If you try this, you have to use water from the 55 to fill the 20 or else you'll probably kill the eggs.
This is what I would do... Forget about the eggs you have now, just leave the situation be. Fill the 20, add your sponge filter and let it run for about 48 hours to get to temp and dechlorinate. Add a little bit of gravel from the 55 to seed it. Leave it mostly bare bottom. Add some furniture for spawning (pots, some floating plants, a couple of larger rocks preferably from the 55 gallon tank so it is covered with beneficial bacteria). Wait a few weeks and move the parents to their new breeder.