We study spin transport in the one- and two-electron regimes of parallel-coupled double quantum dots (DQDs). The DQDs are formed in InAs nanowires by a combination of crystal-phase engineering and electrostatic gating, with an interdot tunnel coupling (t) tunable by one order of magnitude. Large single-particle energy separations (up to 10 meV) and |g^{*}| factors (∼10) enable detailed studies of the B-field-induced transition from a singlet-to-triplet ground state as a function of t. In particular, we investigate how the magnitude of the spin-orbit-induced singlet-triplet anticrossing depends on t. For cases of strong coupling, we find values of 230 μeV for the anticrossing using excited-state spectroscopy. Experimental results are reproduced by calculations based on rate equations and a DQD model including a single orbital in each dot.