Parallel Transport of Geodesic Velocity Vectors

Question

Given a Riemannian manifold $M$ with Riemannian metric $g_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ and distance $d:M\times M\rightarrow\mathbb{R}$ determined by length of minimizing geodesics, letting $a,b,c,d\in M$ be points such that the minimizing geodesics connecting them with each other are unique, denoting the gradient vector of $d^{2}\left(x,y\right)$ taken w.r.t. $x$, by $v_{x}^{y}$, and having parallel transport from $x$ to $y$ along minimizing geodesics denoted by $P_{x}^{y}$, how can one show that $$g_{a}\left(v_{a}^{b},v_{a}^{c}\right)\leq g_{a}\left(P_{d}^{a}v_{d}^{b}+v_{a}^{d},P_{d}^{a}v_{d}^{c}+v_{a}^{d}\right)$$ holds true?