Gary Seitz (and various collaborators over the years) have worked out lots of concrete information about maximal closed subgroups of classical groups and exceptional algebraic groups over an algebraically closed field of prime characteristic. Much of this is used in the study of maximal subgroups of corresponding finite groups of Lie type. Anyway, a basic 1987 paper on classical algebraic groups by Seitz is here.
I don't have all this information at my fingertips, but I suspect you can already draw a complete answer to your question from this substantial paper. He imitates some of the methods used by Dynkin in characteristic 0, though his results are more subtle.
[ADDED] Probably you don't need to get into the rather elaborate inductive framework of the long paper by Seitz, since you are dealing with semisimple groups of rank 2. (Also, it doesn't seem to matter whether you allow characteristic 2 here or what the isogeny type of your groups may be.)
The older 1971 paper of Borel-Tits here already implies that any maximal proper closed (say connected) subgroup of a (connected) reductive group $G$ must be either parabolic or reductive. This is written up in Theorem 30.4(a) in my 1975 Springer text on linear algebraic groups. Parabolic subgroups are easy to describe (up to conjugacy), even if the root system of $G$ isn't irreducible. So for example in your direct product of two copies of a rank 1 group, proper parabolics involve the direct product of a Borel subgroup $B$ in one factor and the entire group in the other factor. To get a maximal proper reductive subgroup, you can use as indicated a diagonal embedding.