Let $(M,g)$ be a Riemannian manifold and $\sigma(t)$ a geodesic on $M$. I'll write $\Pi_{t_0}^{t_1}$ for the parallel transport from $\sigma(t_0)$ to $\sigma(t_1)$ and $[\cdot, \cdot]$ for the Lie bracket on vector fields.
Is it true that $\Pi_{t_0}^{t_1}([X,Y])=[\Pi_{t_0}^{t_1}(X), \Pi_{t_0}^{t_1}(Y)]$ for $X,Y$ vector fields in $M$?
If not, there exists a local frame of $M$ having that property?
For my final purposes, I am interested on veryfing that if $[X,Y]$ vanishes, so does $[\Pi_{t_0}^{t_1}(X), \Pi_{t_0}^{t_1}(Y)]$, but if the property holds in more general settings would be great.
EDITED: As Didier pointed out, $[\Pi_{t_0}^{t_1}(X), \Pi_{t_0}^{t_1}(Y)]$ is not well defined since $\Pi_{t_0}^{t_1}(X)$ is a vector field along $\sigma$ (and possibly not extendable).
So lets try doing the following. Take an arbitrary basis $\{\dot{\sigma}(0),e_1,...,e_n\}$ where $dimM=n+1$. So consider $c: (-\epsilon, \epsilon)\rightarrow M$ a curve such that $\dot{c}(0)=e_i$ and take the variation through geodesics given by $\Gamma(s,t)=exp_{c(s)}(t\xi_{c(s)})$. On this variation define a local frame in the following way: parallel transport $\{\dot{\sigma}(0),e_1,...,e_n\}$ firstly along $c(s)$ and secondly along the main curves $\Gamma_s(t)$. Then we have a local frame defined on the $2$-manifold given by the variation, and we can compute here $[\Pi_{t_0}^{t_1}(e_i), \Pi_{t_0}^{t_1}(e_j)]$ (I know, it is only defined on the variation). So the question in this case is: can I choose a basis $\{\dot{\sigma}(0),e_1,...,e_n\}$ such that both Lie brackets (${e_i,e_j}$ and $[\Pi_{t_0}^{t_1}(e_i), \Pi_{t_0}^{t_1}(e_j)]$) vanish?