First off,
Disneyland in the rain is beautiful, and if it only rains for a couple hours then it can clear out a lot of crowds, make all the sidewalks pretty, and just be something fun and different! I hope you don't cancel your trip because of rain that may or may not even materialize.
My own experience for our trip in December--the forecasts this far out tended to say it would rain the whole time (Wednesday - Sunday). Actually, some of the forecasts predicted rain at last Thursday-Sunday all the way up through the days we actually took our trip.
In reality, it rained some on Friday (not too bad until evening and even then it wasn't super-hard) and pretty solidly through almost all of Saturday. Wednesday was lovely and Thursday and Sunday were partly cloudy, to my recollection. My mom was unhappy that she didn't get to see the sun (she came in Thursday afternoon and left Saturday afternoon) but for my five-day window, 3 days were dry. (I think it rained overnight Thursday night/Friday morning?)
Some surfaces get VERY slippery in the rain, so be careful. A lady in front of us slipped and fell in the downhill area of the queue for Space Mountain.

(She got right up, helped by her family; she appeared to be more embarrassed than hurt.) Outside of some rides, we were told to shake out our umbrellas before walking inside--again, things got quite slippery indoors. Shoes with traction are good.
The Christmas parade was canceled due to rain.

I don't know about the Pixar Play Parade, as it does involve water, so maybe it can go when it's sprinkling? Teacups and Alice in Wonderland were closed. The Mark Twain and Columbia did not run. I think Autopia was closed on Saturday due to rain. I don't think any of the indoor rides were affected at all, but obviously if it's pouring hard and the queue is outdoors, Disney may stop it.
I take it back that the indoor rides weren't affected...There was a fairly large group of people coming in to Innoventions to get out of the rain!

The CM letting us in emphasized about three times how warm and dry it was inside.
Tomorrowland seemed a pretty popular place to be in the rain, since most of the attractions are basically fully indoors.
Finding a place to eat can be tricky; seating was full up at Hungry Bear and I think we waited about 30 minutes in line to get our food. (One person was also saving two tables for people and they came with their food about 2 minutes before we actually finished eating, which didn't help the seating situation in general...but that's another matter.)
Oh, and if it's raining hard you might not want to bother getting Indy FPs--on Saturday when it was raining hard during our trip, they just let EVERYONE into the show building, whether or not you had a FP--I guess they thought it was too mean to make people wait out in the rain? So the FP line was longer than we'd seen it that trip and having a FP appeared to be meaningless. (The line still was not terrible, though; I think we backed up into the spike room, versus the excavation/well area that it usually stopped in.)