The conclusion in this answer is obviously wrong, as shown by your own answer indicating that you did in fact receive a J-1 visa. I'm going to leave this answer as is, because as far as I can tell, it is a correct interpretation of the law and the Foreign Affairs Manual.
This means one of two things:
- I have overlooked or misinterpreted something in the law or the FAM, or
- Actual practice deviates from the law and the FAM.
I hope that anyone reading this will help identify anything I've overlooked or misinterpreted, or shed any other light on this apparent discrepancy. Please add a comment if you can help.
The original answer:
Probably. As the immediate family member of a diplomat in A-1 status, you should only be eligible for admission in A-1 status:
Diplomats and Officials Requiring A-1 Visas - Examples:
- Head of State or Government, regardless of the purpose of travel
- Official coming to serve at a foreign embassy or consulate in the United States, such as an ambassador or consul
- Government minister or cabinet member coming for official activities
- European Union (EU) and African Union (AU) delegation representatives
- Immediate family members of an A-1 visa holder
The last item links to a section on the same page that includes this paragraph:
Immediate family members are defined as the spouse and unmarried sons and daughters who are members of your household, even if studying in a different location. An immediate family member may also be a person who will reside regularly in your household, is not a member of some other household, and is recognized as your immediate family member by the sending Government, as demonstrated by eligibility for rights and benefits, such as the issuance of a diplomatic or official passport or travel and other allowances.
This obviously applies to you unless you are married.
I suppose the first thing you should do is either call the nearest US consulate yourself to ask (or ask your diplomat parent to ask the State Department) whether you can be admitted in J-1 status. I suppose you can't, but maybe I'm wrong.
In A-1 status, you cannot work without express authorization. If you must be admitted in A-1 status, you will probably have to apply for an "employment authorization document" before you can begin the internship. The application is form I-765. The instructions note that the application must be submitted by the Indian mission to the US Department of State:
Submit Form I-765 with Form I-566, Interagency Record of Request-A, G, or NATO Dependent Employment Authorization or Change/Adjustment to/from A, G, or NATO Status, Dependent Employment Authorization, through your diplomatic mission to the Department of State (DOS).
Other relevant regulations are at 8 CFR 214.2(a). Among other things, this mentions that the availability of employment authorization for dependents depends on reciprocity. I have met an Indian who has an EAD as the dependent of a G-visa holder working at the UN, but I don't know whether that implies anything for your situation.
Another relevant regulation is 22 CFR 41.22(b):
An alien entitled to classification under INA section 101(a)(15)(A) shall be classified under this section even if eligible for another nonimmigrant classification.
This is the regulation that requires you to use an A-1 visa if you are eligible for one. There is more discussion of that in the Foreign Affairs Manual at 9 FAM 402.3-4(A).
If you are required to apply for an A-1 visa, the question of nonimmigrant intent disappears, because most grounds of ineligibility, including that one, do not apply to A-1 status. On the other hand, if you can apply for a J-1 visa, you will have to show nonimmigrant intent. But your circumstances do not raise a particular red flag in that regard: you're studying at a foreign university and your parents are not settled in the US.
The State Department's Office of Foreign Missions has a page of frequently asked questions about employment authorization, but if your parents are with India's mission to the UN you should instead consult the analogous page of the US Mission to the United Nations.