  We study the formation and growth of a skyrmion crystal (SkX) phase in a
frustrated antiferromagnetic Heisenberg model on a triangular lattice with
Dzyaloshinskii-Moriya interaction (DMI) in the presence of an external magnetic
field. We build phase diagrams in the temperature-field parameter plane,
featuring first-order phase transitions and tricritical points, and study their
evolution with the increasing DMI strength. It is found that already relatively
small DMI intensity can lead to the appearance of the SkX phase at low
temperatures in the vicinity of the meeting point of the remaining helical,
coplanar up-up-down, and vortical-like ordered phases. By means of a parallel
tempering (PT) Monte Carlo algorithm we find that the minimum value of DMI, at
which the skyrmion phase emerges, $D_t \approx 0.02$, is one order of magnitude
smaller then previously reported. We demonstrate the efficiency of PT in
reliable and precise location of the phase boundaries between different phases
separated by either very large (strong first-order transition) or barely
detectable energy barriers, where the standard Monte Carlo approach may easily
fail to find a stable solution.
