Final answer:
A sentence fragment is identified as a group of words that lacks either a subject or a verb, or both. In this case, option A is a sentence fragment because it lacks a main verb.
Step-by-step explanation:
The sentence fragment in question is option A: My parents' business flourishing, thanks to online advertising. This group of words lacks a main verb and is therefore not a complete sentence. It describes a condition but does not include a subject performing an action, which is a requirement for a complete sentence. To correct this fragment, we would need to add a verb, such as 'is' before 'flourishing' to complete the thought.
Contrastingly, option B: The Riveras make homemade apple cider with their cider press. is a complete sentence because it has a subject, 'The Riveras', and a verb, 'make', forming a complete thought.
The sentence fragment is Option A: My parents' business flourishing, thanks to online advertising. This fragment lacks a subject and a verb. It can be corrected by adding a subject and verb, such as 'My parents' business is flourishing thanks to online advertising.'