It feels absurd to have this occur in the parser, as if you can somehow account for all the cases via guessing what people I mean. You absolutely
must have a grouping operation like {} at least available as a fallback.
I feel like this is a lesson people keep learning over and over across programming languages: don't let your syntax try to infer the intended meaning of what was typed, just do something simple and predictable.
So (b) seems like the only sane choice, but having the grouping operation being e^(abs(x)) is also crazy. You need something like TeX's e^{abs(x)}.
Personally I think the "best" workaround for this is to decouple the text representation and the representation the editing happens in. Allow people to type e^abs(x) and then the editor translates that into e^{abs(x)} while letting you edit it as an exponential, but if you're working on the underlying text file then you have to write e^{abs(x)}. For that matters, you can just have the editor represent it as a literal superscript.
(My hunch is that this is generally much-needed across languages: let the editor do some of the parsing work. I think there's much to be gained by separating the two, because you can keep the parser of the actual language simple while still getting all your editing-efficiency improvements out of the editor).