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We consider mean-field vector spin glasses with self-overlap correction. Thelimit of free energy is known to be the Parisi formula, which is an infimumover matrix-valued paths. We decompose such a path into a Lipschitzmatrix-valued path and the quantile function of a one-dimensional probabilitymeasure. For such a pair, we associate a Parisi PDE generalized for vectorspins. Under mild conditions, we rewrite the Parisi formula in terms ofsolutions of the PDE. Moreover, for each fixed Lipschitz path, the Parisifunctional is strictly convex over probability measures.

Parisi PDE and convexity for vector spins