Sequential Price Competition for Perfect Complements
Just looking at the setup, everything is symmetric. This means that if they were to move simultaneously then we should have $p_1 = p_2$, and the payoffs for the two firms the same.

According to your calculation, $(p^*_1,p^*_2)=(\frac{a}{2},\frac{a}{4})$. Since $Q_1 = Q_2 \Rightarrow \pi_1 &gt; \pi_2$. So it's quite clear that in this case, the first mover has the advantage.

Intuitively, you can figure out which mover has the advantage by simply asking, would the first mover get higher payoff if he/she moves second?

Mathematically, if strategies are strategic substitute, then you can show that the first mover will have the advantage. Reference here.

Added later: one interesting way (I think) to look at this is this is just like Cournot competition, just with price and quantity switched.