See in the official website for both the examination, there were no restrictions for sitting in multiple counsellings. So you can appear for the NATA counselling simultaneously without any doubt. You should keep in mind that through counselling when you choose the college you must be very keen about the course you want to choose. If you are really interested in BARCH then going through NATA will be the best thing for you and you can try out in EAMCET as well in order to have a safe side as well.

To know more about NATA counselling visit the link below:

https://engineering.careers360.com/articles/nata-counselling

To know more about EAMCET visit the link below:

https://engineering.careers360.com/articles/ap-eamcet-counselling

I hope this answer helps. All the very best for your future endeavors!