Can I move my sink drain connection to the parallel vent pipe?
I can’t speak to a specific code or have an alternative plan in mind, but I think you would have three problems with your proposed drawing.

First, if there is a blockage between the first and second floors, there is no longer venting for the first floor sink.

Second, if there is a blockage just below the first floor sink and the second floor is not aware of it, they’re water will back up into the first floor sink.

Third, first floor could end up sending some of their vent gas into the second floor sink depending on the path of least resistance.