Which phrase I should use?
"I told them I will pick it up on Sunday morning" or "I told them that I will be picking it up on Sunday morning"
The second one is possible if you need to say how you will be spending your Sunday. The first one is what is expected if you need to say when you will pick it up.
A: They need it on Monday. Will that be okay?
B: Yes, I've just told them I will pick it up on Sunday morning.
A: They are hoping you can join them for breakfast on Sunday.
B: Well, they should know I can't. I told them I would be picking it up on Sunday morning. (This makes sense only if right before this you have been discussing what "it" is.)
You CAN say the second version as the answer to the first, but it's not necessary.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.