Let's break the sentence into parts:
Sentence structure: [subject] + [まで] + [noun] + [を] + [ru-verb] + [わけ] + [に] + [はいけない (changed to はいかない)]
自分 (jibun) = [noun] oneself (including myself), this subject has similar trait like 私 has
まで (made) = [particle] until, up to, even (when used after a subject)
父親 (chichi oya) = [noun] father
を (o) = direct object/straightforward particle
見捨てる (misuteru) = [transitive -ru verb] abandon, leave
わけにはいかない (wake ni haikanai) = [particle] strong (or "extreme") version of ーてはいけない used with -ru verb, means something that must not be done at all
Note that "まで" in "自分まで" here doesn't used to mention "until myself", it's more proper with "even myself", or just simpler "myself". So that the sentence literally have this meaning:
(Even) myself must not leave my father (at all times).
To fit with situation in entire sentence, it modified to this form:
I can't abandon my father (at all).