Does parallel transporting require an ambient space? Can someone summarize why an ambient space isn't needed to measure curvature when parallel transporting tangent vectors or vector fields along a curve on a Riemannian manifold? How do we define the vector's direction and magnitude without one? 
 A: You do not need the ambient space because formally Riemannian manifolds are defined via abstract glueings of Euclidean spaces, which you can think as little disks or cubes if you like. A human being living in a submanifold of the ambient space will not be able to "feel" the ambient space's existence. Rather to tell the distance and flatness in his/her space this person has to rely on the metric. 
In particular, we cannot measure a "vector". There is no "vector" in a sphere like earth, for example. What you can measure is the tangent vector that is tangent to some point in the sphere, and the length of it using the metric at that point by
$$
|v|=\sqrt{\sum^{n}_{i=1,j=1} g_{ij}v_{i}v_{j}}
$$
Similarly you can "feel" the angle between one tangent vector to the other at that point by
$$
\cos(\theta)=\frac{|u\cdot v|}{|u||v|},|u\cdot v|=\sqrt{\sum^{n}_{i=1,j=1}g_{ij}u_{i}v_{j}}
$$
I hope this addressed the issue of vectors. To measure curvature (I assume you either mean the Riemann curvature tensor or the sectional curvature), you need to compute the Levi-Civita connection based on the metric. In other words, once the metric is fixed you basically "know all you want to know" about the manifold, and there is no need to introduce some arbitrarily ambient space like when we defined the concept of the submanifold. 
A: It is an intrinsic concept as given a covariant derivative $\nabla$, the parallel transport along a curve $\gamma$; is obtained by integrating the condition $\scriptstyle{\nabla_{\dot{\gamma}}=0}$. 
Conversely, if a suitable notion of parallel transport is available, then a corresponding connection can be obtained by differentiation:
Consider an assignment to each curve γ in the manifold a collection of mappings
$$\Gamma(\gamma)_s^t : E_{\gamma(s)} \rightarrow E_{\gamma(t)}$$
such that
$$\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - V_{\gamma(0)}}{h} = \left.\frac{d}{dt}\Gamma(\gamma)_t^0V_{\gamma(t)}\right|_{t=0}.$$
A: I guess I don't really understand what you mean by the orientation of a vector in this context. The parallel translation of the vector $X_0$ from the point $p$ to the point $q$ is the vector $X(1),$ where $X(t)$ is the unique solution of the linear system of equations 
$$\nabla_{\gamma'(t)} X = 0$$ with initial condition $X(0) = X_0.$ There is no mention of 
an ambient space in this definition.  
