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 ($sim$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 $mu$eV 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.
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