Consider the partition with maximizing the number of pairs that one of them from each bus. Suppose the number of pairs in each of two buses are $a,b$ and pairs between buses is $c$. Note that for each student $S$ in the first bus, the number of students in the second bus who is friend with $S$ is at least that in the first bus. Summing this for all students $S$ in the first bus give us $c\geq 2a$. Note that $a+b+c=k$. If $a>\frac{k}{3}$, we have $c>\frac{2k}{3}\Rightarrow a+b<\frac{k}{3}<a$, impossible. So $a\leq \frac{k}{3}$, similarly $b\leq \frac{k}{3}$.

It is interesting to ask what is the minimum value of $\max_{i\in \{ 1,2\}}{ \{ \text{the number of pair(s) in bus number } i\} }$ in term of $k$.