Part Time

Yes, you can do the same as it will improve your mindset and skills.

I am not sure a part-time degree given in core.engimeering domains by anyone.

As per supreme court rules, you cannot do two degrees at the same time from same or different universities. Unless your MBA is not a degree and just a diploma. Then, in this case, you can do LLB from Mumbai University.

For long term career growth, it will be ideal if you go for two-year full-time regular MBA from any of the top 50 B-Schools. In case you can not take out two years for full-time MBA due to personal constraints then you may opt for regular one-year EMBA. Your third option should be part-time MBA and last option is MBA through distance learning.

Fleximode MBA means a correspondence or a part-time MBA. So, whichever university you are applying which offers a fleximode then check their registration under University Grants Commission (UGC).
Also, as a piece of advise please do a full time MBA if you have the time as that will give you proper insights into the management degree and recruiters prefer them over part-time MBA.

You can do a part-time job while doing BCA degree. BCA degree has a big syllabus and to become a good software engineer, you require to put a lot of effort. Therefore, you must evaluate whether you can handle both studies and job pressure simultaneously.