The answer to your question would be that the sentence that lacks parallel structure is the following one: I am going to go to the store, shopping for ingredients, then start cooking will begin. That is, your answer would be D.
Parallel structure, aka parallelism, is the repetition of a grammatical structure within a sentence. In D, the speaker states that he/she is going to go to the store, that is, the first verb is "future going to". Therefore, all the other verbs should follow the pattern going to+infinitive. However, this is not the case since one of the verbs is in the present continuous form (shopping) and the other one is followed by another verb phrase (start cooking will begin), making the sentence ungrammatical. For the sentence to be parallel it should be as follows:
a) I am going to go the store, shop for ingredients, then start cooking.