Does rain itself affect barometric pressure?
Under a hydrostatic assumption the basic equation is p=/rho R T. So under this easy equation the pressure depends to the air density and the air temperature. As you may know diabatic processes during rain yield to a decreasing temperature. but the magnitude in change of pressure against the synoptic scale forcing is low. 

Please be aware of mixing stuff together answering this questions. The question was not to answer how precipitation is forming. It was just the question about the rain influencing the pressure.