Thoughts,
1) Double check again to see if your radiator is full. (Perhpas just refilling and moving the truck was enought for the head gaskets to leak.) You also could put a radiator tester on the radiator and start the truck. (On and off first. Then on for longer.) Looking for pressure quickly building in the radiator from compression gasses.
2) If you had a cavitaiton hole it could leak while sitting. (Not likely, just possible.)
3) I would check but go ahead and replace the CDR valve and the rubber gromet it plugs into. The CDR is one of the most overlooked maintenance issues for these trucks.(You really do not know when it has gone bad, cleaning during oil changes, and replacing every so often is the best way to avoid it failing.) A bad one can cause a blown head gasket. (Let's too much oil into the intake manifold.) Many mechanics and truck owners do not replace the rubber boot that seals the top of the engine to the CDR valve, when replacing the CDR valve. So these boots crack and fall apart, letting blow by out but also dust and in this case perhaps water in. Then add rain running down the front windshield, waterfalling off the body, and down on top of your engine (About where the CDR is.) and running into your motor via the bad CDR boot on the top plate. (But it sounds like a lot of water.)
4) Perhaps also look for a bad turbo intake boot. I do not have a turbo, but anywhere your turbo intake goes from metal to rubber could be a spot for intrusion.
5) Last look at the top plate (Under injector lines), it is pretty thin as I recall. Add years of use, some abuse, some salt from the roads, water sitting on it every rain, maybe your top plate has rusted through and lets water pour into the lifter valley. Maybe do a smoke test from the oil fill hole to see what happens.
Let us know what the resolve is!