The query you posted is a common problem faced by every another students at your phase.
Student get confused between Science and mathematics, if chosen Science the to pursue regular graduation -post graduation or Medical.
The main factor to be considered is your Interest and choice. What you want to pursue honestly? Candidates are doing great in both the field- MBBS as well as Biotech sector. There is nothing good or bad between them.
But you have to realize your own capability in doing better in any of them.
Can you analyze the reason for why you may not get chance for admission in any good college even after dropping one year for NEET? If you decide to pursue MBBS, you have to overcome your failure rather than trying to get short cuts for success. Escaping can never lead to success.
Your parents are right in saying that once you will enroll yourself for any course, you will not be able to focus on your preparations for NEET. So, if you really want to crack NEET with good scores, follow what your parents are saying because Kota (Rajasthan) is famous for coaching centers focused on engineering and MBBS examinations. But these are costly so, you may consider you financial ability for that.
Living & Learning- simultaneous processes!