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 pp 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 → ℓℓbb, WH → ℓνbb, and ZH → ννbb 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 are 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).