  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~fb$^{-1}$ of
$p\bar{p}$ collisions at $\sqrt{s} = $ 1.96~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 $ZH\rightarrow \ell\ell
b\bar{b}$, $WH\rightarrow \ell\nu b\bar{b}$, and $ZH\rightarrow \nu\nu
b\bar{b}$ final states, we compare the standard model (SM) Higgs boson
prediction, $J^{P}=0^{+}$, with two alternative hypotheses, $J^{P}=0^{-}$ 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 $J^{P}=0^{-}$ and
$J^{P}=2^{+}$ hypotheses at the 97.6$\%$ CL and at the 99.0$\%$ CL,
respectively. The expected exclusion sensitivity for a $J^{P}=0^{-}$
($J^{P}=2^{+}$) state is at the 99.86$\%$ (99.94$\%$) CL. Under the hypothesis
that our data is the result of a combination of the SM-like Higgs boson and
either a $J^{P}=0^{-}$ or a $J^{P}=2^{+}$ signal, we exclude a $J^{P}=0^{-}$
fraction above 0.80 and a $J^{P}=2^{+}$ fraction above 0.67 at the 95$\%$ CL.
The expected exclusion covers $J^{P}=0^{-}$ ($J^{P}=2^{+}$) fractions above
0.54 (0.47).
