It really depends on the weather... I have seen parades happen when the weather was threatening rain. If it is raining, they usually do the rainy day cavalcade... omnibus with characters on it, horseless carriage with Mickey and Minnie... you get the picture.
<3 There are a million cupcakes in the world but only one is Cupcake Terror. <3
Remember, the call for full or Rainy Day parade has to be made early. It takes time to get all the floats out or to put them back in. That's why sometimes you will see a Rainy Day parade when it is sunny - if it had been raining earlier the call would have already been made.
I was in Tokyo when it rained very hard for a very short amount of time. That was enough to alter the day parade into the rain version. The parade came out and the characters were wearing rain gear. We all were dying in the 90+ heat and blinding sun.
They'll run it if it looks like rain. And if it's sprinkling they will often send the parade out for just a run down Main Street with no choreography, just waving and smiling. They usually have the parade waiting at the entrance of the route and make a final decision last minute about whether it will run full out or not.
If it's pouring, they definitely won't send the parade out.
There I was somewhere West of Tomorrowland
Flying through the stars just like Peter Pan
I said...Dang this looks too good to be true...
It's a nice day when you wake up in Disneyland...
-Five For Fighting, "Disneyland"
They send POD out with choreography and no show stops sometimes. It's not the floats that are the problem. They'll get rained on backstage (if there's already been one parade) anyways. The problem is the costuming and makeup just goes to heck if rained on.
Thus, you may spot the Omnibus, or both them, crossin Main Street to pick up the characters.
When i was there on Jan 4th, the second showing of ACF ran in the rain, with no ground performers and the floats were paced close together, a couple had to be towed, one on the turn from hub to matterhorn way, and one the on the hill and turn up by Alice. My friend was Valeting Toy Factory that day. He said that they had to get the units from Egghouse to Warehouse, so they wouldnt get water damage. It started to rain that day after the first showing. I have a pic of them covering the units backstage in Egghouse. So if it rains and that happens they usually will run the parade on what they call a "Drive through". No ground performers, fast paced, and limited characters.