Hello!
Basically, I´m with George about his answer,but it isn´t fine at all.
Mainly, "at the same time" means "al mismo tiempo".Like in that example:
"Yo estaba cocinando al mismo tiempo que escuchaba la radio"
(I was cooking at the same time than I was listening to the radio)
It´s the same with "a su vez" But ," a su vez", also implies an independent
course of action,related,inside the same time ratio.So, taking the example before...
"Yo estaba cocinando, mientras que, a su vez, mi hermano preparaba la mesa"
(I was cooking whilst my bro was dishing the table at same time)
Can u see it?
We could say than , "a su vez" = por su parte+a la vez"(by his side+at same time), OK?
I Hope had been helpful. Bye!