DLP has very little rides which are only outdoor, Big Thunder Mountain, Indiana Jones, RC racer, Pays des Conte Fees, Casey Junior and the shows might be effected. Things like the river boat usually stay open.
Just get a poncho and keep going, rain is great to keep the locals away.
It rained lightly the first few hours when I was there last summer. BTM remained open. The Indiana Jones coaster was closed. We just spent a lot of that time on the indoor attractions.
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BTM was open for us when it was raining, but I can't say I'd recommend riding it then. Being pelted in the face by raindrops isn't entirely pleasant. We did ride Indiana Jones, too, and the night shows still ran. The walkways behind the Main Street shops are a great place to escape for a bit and wait for Illuminations.
Luckily, Paris was built with rain in mind... If you have a look around youtube, you'll find a handful of videos showing the so called "covered walkway" that gets you from Main Street round to (I think) at least Adventureland in way that's almost entirely covered from the rain. There's a few other (smaller) walkways to get you around other bits of the park in the dry as well.
Also, as a local, I would recommend wearing shoes and outfits designed for the rain. The DLP ponchos are expensive and in no way confortable to wear!
Rain coats are already packed.... it’s the rain shoes I’ve been debating....
I would also recommend taking warmer
clothes. Rain in Paris is not like rain in Florida - especially now it’s quite chilly - like my feet would be numb if I were in sandals today...