Aspects of this question have been thoroughly treated in The Classification of Finite Simple Groups, Number 3, by Gorenstein-Lyons-Solomon, AMS, 1994: see especially their Table 3.3.1 for the Chevalley groups. In your notation (which differs somewhat from theirs), the $p$-rank is $16a$. This gives only the rank of a maximal elementary abelian $p$-subgroup, however.
P.S. Concerning maximal abelian $p$-subgroups of $E_6$, the relevant table in Vdovin's thesis (linked by Nick Gill) seems to give the same answer $p^{16a}$.