Will a tarp over the deck allow me to paint the deck during a Seattle winter?
I have made temp tarp roofs over several work sites when building a deck in winter. (Vancouver, BC here, so similar winters).

You should use a new tarp, have a slope for drainage and make sure it is tight and secure.

If any leaks or drips get on the wood it takes longer to dry out in the winter (when it's wet out in general).

I have never painted a deck though, only stains.

The most critical reason for using a tarp roof was keeping plywood dry when we were doing a waterproof vinyl membrane. Most other times they weren't crucial and just for keeping work site relatively dry.