We present constraints on models containing non-standard-model values for the spin $J$ and parity $P$ of the Higgs boson $H$ in up to $9.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ collected with the D0 detector at the Fermilab Tevatron Collider. These are the first studies of Higgs boson ${J}^{P}$ with fermions in the final state. In the $ZH\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\ell}b\overline{b}$, $WH\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\nu}b\overline{b}$, and $ZH\ensuremath{\rightarrow}\ensuremath{\nu}\ensuremath{\nu}b\overline{b}$ final states, we compare the standard model (SM) Higgs boson prediction, ${J}^{P}={0}^{+}$, with two alternative hypotheses, ${J}^{P}={0}^{\ensuremath{-}}$ and ${J}^{P}={2}^{+}$. We use a likelihood ratio to quantify the degree to which our data are incompatible with non-SM ${J}^{P}$ predictions for a range of possible production rates. Assuming that the production rate in the signal models considered is equal to the SM prediction, we reject the ${J}^{P}={0}^{\ensuremath{-}}$ and ${J}^{P}={2}^{+}$ hypotheses at the 97.6% CL and at the 99.0% CL, respectively. The expected exclusion sensitivity for a ${J}^{P}={0}^{\ensuremath{-}}$ (${J}^{P}={2}^{+}$) state is at the 99.86% (99.94%) CL. Under the hypothesis that our data are the result of a combination of the SM-like Higgs boson and either a ${J}^{P}={0}^{\ensuremath{-}}$ or a ${J}^{P}={2}^{+}$ signal, we exclude a ${J}^{P}={0}^{\ensuremath{-}}$ fraction above 0.80 and a ${J}^{P}={2}^{+}$ fraction above 0.67 at the 95% CL. The expected exclusion covers ${J}^{P}={0}^{\ensuremath{-}}$ (${J}^{P}={2}^{+}$) fractions above 0.54 (0.47).