The weather in tropical biomes remains the same, even when the seasons change. So jungles will still rain in the dry season (mid_summer) and savannas will be dry in the wet season (mid_winter):

Forge: 14.23.3.2676
Serene Seasons: 1.1.4
Minecraft: 1.12.2

My mistake marking this as a bug, when it's actually not.

So, rain is only actually affected in the mid dry/wet season. The chart for it looks like this:

So it only rains in desert biomes in late winter/early spring, and rain is only prevented in jungle biomes in late summer/early autumn. I'll update the wiki pages for the tropical seasons to make this more clear.