The only thing I can think of would be that somehow the transmission got into two gears at the same time or more likely a tooth broke off in the differential and is blocked when it tried to mesh.

FWIW, the parking brake would not bring you to a complete, instant stop. I tried one time to see at the speed of about 30 miles per hour and it took about a hundred yards to come to a complete stop using only the parking brake.

That is what I am trying to do is see if the tranny is in fact in neutral.  Haven't had much luck with it.  I was going to take a look at the rear end.  I heard they have C Clamp in there that sometimes will fall off and get stuck in the gears.  Makes you mad when it was running so good too....Thanks for your input!

You could disconnect the drive shaft to determine which end is locked up.

I noticed that when we try to put the transmission in Reverse the rod/arm is not moving at all.  So I disconnected the the drive shaft and thought well at least I can turn the wheels - WRONG.  They are frozen.  I then tried starting it with out the drive shaft hooked up and the same ole thing it will bog down and die.  So I believe the transmission was caught in Reverse and another gear and then that caused the rear end to break.  So I will be out of a lot of money trying to find both of these.  I will just get some used ones.  Thanks again for your help!

So with the drive shaft disconnected the rear wheels would still not turn?

They will not turn?  The other strange thing is that it will still bog down with the engine running and no drive shaft.  It did appear to have 1st, 2nd, 3rd and 4th.  But when you put it in reverse the shifting plate is not moving.  I hope I don't have to purchase a rear end and transmission. 

First, verify that the transmission is in neutral. Use a wrench on the linkages underneath, and position each shaft in the center of it's travel. If you get the front two shafts in the center, even if the trans is stuck in reverse, it should no longer bog the engine down. Be careful tho, and do not rev it up without the driveshaft connected to the trans yoke.

For the sound of it, you have two separate issues, but for both the trans AND diff to lock up at the same time is just too weird. Were there any strange noises before this all happened?

Update on this Corvette.  I have taken the 1/2 shafts off of differential.  I can now turn the wheel hub.  When I grab differential you still can't make it budge.  I have 5 bolts to go to drop it down to see what is going on in the rear end.  I am wonder if the axle locking pin has slipped off and gotten into the gears.  Well anyway I will tackle the transmission after I get this differential turning.  

Yes, you need to check the shifter levers on the tranny.  Make sure as Adam said, that the levers are all pointing straight up and not in a forward or aft detent.