We investigate the interplay between unconventional magnetism and superconductivity in a model of a p-wave magnet on a square lattice. Using a self-consistent Bogoliubov-de-Gennes approach, we analyze the pairing amplitudes, competition, and coexistence of spin-singlet s-wave and spin-triplet p-wave pairings in the presence of a magnetic texture with a helical structure along the x direction that is repeated in the y direction. We find that the magnetic helix selectively stabilizes different pairing symmetries depending on its orientation and strength. In particular, mixed-spin pₓ-wave pairing is enhanced at intermediate magnetic couplings and equal-spin p_y-wave pairing is robust and insensitive to all coupling intensities. When the multiple order parameters are simultaneously considered, we find regimes of coexistence and competition. Increasing the magnetic coupling drives two quantum phase transitions. The first from dominant spin-singlet s-wave to mixed-spin triplet pₓ-wave pairings in a regime of coexistence. The second from spin-singlet s-wave and mixed-spin triplet p-wave pairings with total spin projection S_z=0 to dominant equal-spin triplet p_y-wave pairings with S_z=±1 in a regime of mutually exclusive superconducting phases. Our results demonstrate that p-wave magnetic order does not merely diminish spin-singlet s-wave superconductivity but can actively promote and stabilize spin-triplet p-wave pairing, both intrinsically and in proximity to spin-singlet s-wave superconductors. These findings highlight unconventional magnets as promising materials for realizing robust triplet superconductivity.