  The parallel-magnetic-field $H_\parallel$ dependence of the superconducting
transition temperature $T_c$ is studied for ultrathin films of In, Bi, and Al
grown on GaAs(110). In the case of In films in the monolayer regime, $T_c$
exhibits a quadratic-like $H_\parallel$ dependence, which is one order of
magnitude stronger than that previously observed in monolayer Pb films by the
present authors [Phys. Rev. Lett. 111, 057005 (2013)]. The results are well
reproduced by a model developed for an inhomogeneous two-dimensional
superconducting state in the presence of a moderate Rashba spin-orbit
interaction. The Rashba spin splitting is estimated to be 0.04 eV, which is
much smaller than that expected for monolayer Pb films. In a few-monolayer Bi
film, the suppression of $T_c$ with increasing $H_\parallel$ is comparable to
that in monolayer Pb films. On the other hand, much stronger suppression, which
is attributed to the Pauli paramagnetic effect, was observed for the Al film.
