Why do you want to do that? What do you mean by disorder here? What are those processes being read and output so important that prevents you using an interrupt to handle another task, buttons, etc... Even intel processor slices processing time among different processes, it also uses interrupts to handle the mouce...
As long as you have a single CPU/core, you will never have parallel processing... When you say "disorder" it means that your program is being executed by diffrerent CPUs/cores such that tasks/threads don NOT interfer with each other, which is another story... or that you don not understand how the interrupt is working... Interrupts don not disorder program execution, they redirect it somewhere else that you specify, and then return back to the next instruction that was supposed to be executed if there was no interrupt...
First of all, you should have read resources about handling buttons.. Your diagram implies that you have no idea... You might have found a readily available routine somewhere... or you are still a beginner...
Now introduce to other members what exactly on your mind buddy.. so they can help you further... forget the term "parallel processing" for one AVR chip

You might want to disable the interrupts when you read/output data, this is a wise thing to do inorder not to miss your data, but don not disable them for a long time, then you will miss button pressing events.