Question

What sub-atomic particle is counted to learn the atomic number of that element?

Answer 1

Protons.
They increase accordingly with atomic number

Answer 2

That would be protons.
Subatomic particles are electrons, protons and neutrons.
The atomic number is the number of protons of each atom.
The atomic mass is the sum of number of protons and neutrons.
For example carbon has atomic number of 6 and atomic masa of 12. So it has 6 protons and 6 neutrons.