  We consider a theoretical problem of the upper critical magnetic field
parallel to a conducting axis of a quasi-one-dimensional layered
superconductor. We show that the orbital effects against superconductivity in a
magnetic field are capable of destroying the superconducting phase at low
temperatures if the interplane distance is less than the corresponding
coherence length. Applications of our results to the recent experiments,
performed in the superconductor Li$_{0.9}$Mo$_6$O$_{17}$ [J.-F. Mercure et al.,
Phys. Rev. Lett. {\bf 108}, 187003 (2012)], provide strong arguments in favor
of a triplet superconducting pairing in this quasi-one-dimensional layered
conductor.
