No, unless you leave the lid open for too long.. But I do wrap the plates in sanitized alu foil during growth & storage which is much the same as Parafilm, although less tight. If you leave those upside down without any cover so yes, eventually you will get some contaminations through the gap between lid and the plate. Parafilm is good if you need to store very long cause it will prevent plates from drying out, too. I have only kept the plates for a week or two and then replate or thaw fresh yeast from the freezer. An additional method would be to store the plates inside a sanitized plastic container.
honestly it was always yeast that infected my myclium....
do you use a Flow hood or still air box for the plates? or is yeast just easier to grow than mushrooms?