The hard-gluonic contribution to the first moment of the polarized proton structure function $g_1^p(x)$ is dependent of the factorization convention chosen in defining the quark spin density and the hard cross section for photon-gluon scattering. Two extremes of interest, namely gauge-invariant and chiral-invariant factorization schemes, are considered. We show that in order to satisfy the positivity constraint for sea and gluon polarizations, the polarized valence quark distributions should fully account for the observed $g_1^p(x)$ at $x\gsim 0.2\,$. This together with the first-moment and perturbative QCD constraints puts a pertinent restriction on the shape of $Δu_v(x)$ and $Δd_v(x)$. The spin-dependent sea distribution in the gauge-invariant factorization scheme is extracted from the data of $g_1^p(x)$. It is shown in the chiral invariant scheme that it is possible to interpret the $g_1^p(x)$ data with anomalous gluonic contributions, yet a best least $χ^2$ fit to the data implies a gluon spin distribution which violates the positivity condition $|ΔG(x)|\leq G(x)$. We then propose a more realistic set of parton spin distributions with sea polarization and with a moderate value of $ΔG$. The polarized parton distributions in this work are presented in the next-to-leading order of QCD at the scale $Q^2=10\,{\rm GeV}^2\,$. Predictions for the polarized structure functions $g_1^n(x)$ of the neutron and $g_1^d(x)$ of the deuteron are given.

17 pages, 9 figures. Main changes: (i) all polarized parton distributions are parametrized and determined in the MS scheme, (ii) all figures are modified