What can I use to set the strawberries on to cool if I don't have parchment paper or a cooling rack?
If you are glazing only a few strawberries you can pick up the strawberries with a (wooden) skewer (toothpicks might be too short) and then poke the other end into styrofoam, a sponge or anything similar or put these into a tall and heavy glass.
Just put a dinner plate in the fridge, then set the strawbs on that... Make sure the strawberries are drained off well after you dip 'em, and then the chill of the plate will set the chocolate.
If you have stone countertops you could set them straight on to them as well...
Wax paper or aluminum foil should substitute well for parchment in this case.