The correct answer is B. He promised to love her in sickness, in health, in poverty, and in wealth.
A parallel structure refers to a sentence that contains the same form of a word or phrase throughout. If you look at sentence B, the parallelism can be observed in prepositional phrases in sickness, in health, in poverty, and in wealth - all of these phrases contain the preposition IN and a noun.
Sentence B is also correct regarding commas - there shouldn't be a comma immediately after the pronoun HER.