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Generalizations of the parallelogram law for the Schatten 2-norms have been given in the following form: If $\\mathbf{A}=\\{A_1,A_2,...,A_n\\}$ and $\\mathbf{B}=\\{B_1,B_2,...,B_n\\}$ are two sets of operators in $\\mathcal{C}_2$, then $$\\sum_{i,j=1}^n\\|A_i-A_j\\|_2^2 + \\sum_{i,j=1}^n\\|B_i-B_j\\|_2^2 = 2\\sum_{i,j=1}^n\\|A_i-B_j\\|_2^2 - 2\\Norm{\\sum_{i=1}^n(A_i-B_i)}_2^2.$$ In this paper, we give generalizations of this as pairs of inequalities for Schatten $p$-norms, which hold for certain values of $p$ and reduce to the equality above for $p=2$. 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