We can pave in the rain, however we do not like to. If your paving project is located more than a 20ish minutes away from an asphalt plant, I would wait for dry weather. Asphalt comes to us in dump trucks and is very hot when it leaves the asphalt plant. Even tarping a load of asphalt will not prevent a load from getting cold for too long. Paving on cold days is bad enough, then add the rain and it is terrible. The rain cools the asphalt much faster and makes joints look bad.
Passing showers or sprinkles can be dealt with but also is not ideal. If we know that there is a chance we could be dealing with rain we will plan accordingly. Sometimes we will schedule trucks to be on standby at the asphalt plant until we are ready for them. With the asphalt still in the silo it will stay hot much longer.
Overlays, however should never be paved in the rain. If a paving contractor tries to do a asphalt overlay in the rain it will almost always fail. Tack (basically asphalt glue) will not set up and be worthless when trying to bind the two layers of asphalt.