No. You may get some reduced spiders or bees from the pairing as well as reduced normals...but all of the reduced pattern would come from your male, not from the spider or bee (unless they had reduced pattern in their ancestry). The spider gene reduces pattern as well, but only in animals that actually have that gene. It's one gene, it does not somehow 'leak' any of its characteristics into animals that don't have the gene.

Reduced pattern spiders look pretty neat.