It's the other way around. This sentence means "I think I have to thank my father and mother." ｘｘｘ（し）なくてはいけない　=　xxx（し）なければならない　=　xxx（し）なきゃいけない　=　have to do In terms of morphology, しなくてはいけない reads "it is no good if somebody does not do something."

iku = to go ikeru = can go (potential form) ikenai = can not go in idiomatic usage ikenai basically means won't go well. suru = to do shinai = not do shinakute = the gerund form of shinai, you use this form to create compound verb phrases. X shinakute wa ikenai = not do X, won't go well (implicitly if you don't do X, things won't go well) an alternative phrasing would be X shinai to ikenai. If you don't do X, then things won't go well.