Your reasoning is good, except for one thing: it doesn't account for the use of "even," which wants to introduce a grammatical structure of the same type -- in this case, another finite verb phrase in the simple present. I would add "and":
2a) Sometimes he looks right at you, talks to you, and even asks how you are, but he never remembers it in the morning.
Your interpretation would work with "even" if the participial modifying phrase were a member of a series or at least a pair of coordinated items. I would prefer a series. Thus you could have:
3) Sometimes he does very strange things, looking right at you, talking to you, and even asking you how you are.
So, yes, parallelism is of the essence here. But it isn't parallelism for parallelism's sake. The grammatical parallelism is mandated by the use of "even." You can have the parallelism at the level of the finite verb phrase or at the modifier level.
Your student deserves a cupcake.