What follows is a discussion in "low level". For "high level" discussion go to connections summary.
1. In an abstract setting of a $n$-dimensional manifold $M$, given a parallel transport $\tau$ we can construct a covariant derivative operator $\nabla$. See the sketch of the construction at parallel transport#Covariant derivative, that can be generalized to any manifold, not only surfaces.
2. Conversely, given a covariant derivative operator $\nabla$ we can construct a parallel transport $\tau$. See this construction.
3. A Riemannian metric $g$ on $M$ gives rises to several covariant derivatives (or parallel transports). There is only one which is torsion free, the Levi-Civita connection $\nabla_g$, with parallel transport $\tau_g$.
4. In $\mathbb{R}^N$ the standard metric $g_{std}$ gives rise to the standard covariant derivative $\nabla_{std}$ which corresponds to a parallel transport $\tau_{std}$ that is the usual absolute parallelism of $\mathbb{R}^3$.
5. If the manifold $M$ is a surface immersed in $\mathbb{R}^3$ (and I guess this is valid for any manifold immersed in $\mathbb{R}^N$) we can construct a metric induced by the "ambient metric", $g_{ind}$. This inherited metric gives rise, according to point 3 above, to $\nabla_{ind}$ and $\tau_{ind}$.
6. In particular, $\tau_{ind}$ can be constructed from $g_{ind}$ by means of geodesics and holonomy. See parallel transport#Intrinsic construction for surfaces. For 3-manifolds @needham2021visual provides three constructions in page 282.
7. But $\tau_{ind}$ can be constructed directly by means of translation and projection in $\mathbb{R}^N$, that is, by using $\tau_{std}$ and $g_{std}$. This correspond to parallel transport#Extrinsic construction.
8. Respect to $\nabla_{ind}$, it can be constructed from $\tau_{ind}$, according to point 1 above, but also from $\nabla_{std}$ and $g_{std}$ (see linear connection#How to get one).
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Author of the notes: Antonio J. Pan-Collantes
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