Can one pursue multiple degrees via distance education mode in India?
The Department of Higher Education, Ministry of Human Development, Government of India says,

It is up to the concerned academic institution/university to recognize the qualification including certificate diploma, degree, etc. for the purpose of academic pursuit, i.e. continuing education for acquiring another academic qualification, with it. As regards recognition of academic qualifications for the purpose of employment, it is the prerogative of the concerned employer to take a view on the recognition of the degree, diploma, etc
(Last Updated by admin on Tuesday, 19 April 2016 - 12:27pm)

My interpretation of this is that it is up to individual institution/employer to recognize the multiple degrees.
According to Times of India Feb 2, 2013

According to the decision of the DEC communicated to UPRTOU. vice-chancellor Prof AK Bakshi, while a student cannot pursue two degree programmes simultaneously, he can do the same through distance or combination of distance and regular modes from the same or different university institution in various combinations.

So, I think the answer is, yes you can.
However, please double check this with the government in India so you can get an official answer. Please do not trust the strangers on Internet. Instead, trust your government.