Let $M$ be an $n$ dimensional connected parallelizable manifold and $\{X_1,\dots,X_n\}$ a basis for the tangent bundle. Consider the following
Claim. There existis $p\in M$ such that for all $q\in M$ there exists $(a_1,\dots,a_n)\in\mathbb R^n$ such that there is an integral curve $\gamma$ of $\sum_{i=1}^na_iX_i$ that satisfies $\gamma(0)=p$ and $\gamma(1)=q$.
That would allow us define a map for some $U\subseteq\mathbb R^n$, $U\hookrightarrow\mspace{-15mu}\rightarrow M$ that works as a "canonical chart" given the basis and the point, mapping each $a\in U$ to the endpoint of the corresponding integral curve. The term is in quotes because $U$ might not be necessarily open.
This is quite easy to see in connected Lie groups for example, where $p=1$ due to group-algebra correspondence. Can we extend it to all parallelizable manifolds? If not, what extra structure do we need?
EDIT: Inspired by Mariano's comment, here's a quick example of this construction. A nonzero vector field $X$ in $\mathbb R$ corresponds to a strictly (w.l.o.g.) increasing $C^1$ function $f:\mathbb R\rightarrow\mathbb R$ via $X=x\mapsto(1/f'(x))\partial$. The integral curves of $aX$ then all satisfy $f(\gamma(1))-f(\gamma(0))=a$. Fixing any $p\in\mathbb R$, our chart is then a diffeomorphism between $\mathbb R$ and $(f(-\infty)-f(p),f(\infty)-f(p))$.