Does there exist a non-parallelisable manifold with exactly $k$ linearly independent vector fields? 
For any positive integer $k$, is there a smooth, closed, non-parallelisable  manifold $M$ such that the maximum number of linearly independent vector fields on $M$ is $k$?

Note that any such $M$, for any $k$, must have Euler characteristic zero by the Poincaré-Hopf Theorem.
Without the non-parallelisable hypothesis, the $k$-dimensional torus is an easy example.
One might think that the product of a $k$-dimensional torus with a manifold which admits no nowhere-zero vector fields provides an example, but this doesn't necessarily work. For example, the product of a torus and an even dimensional-sphere is actually parallelisable. The reason this approach fails with the even-dimensional sphere is that it is has stably trivial tangent bundle; maybe using a manifold with non-stably trivial tangent bundle might work.
Determining the maximal number of linearly independent vector fields on a manifold is not easy in general. Even for spheres, the answer is complicated: if $n + 1 = 2^{4a+b}u$ where $a \geq 0$, $0 \leq b \leq 3$ and $u$ is odd, then the maximal number of linearly independent vector fields on $S^n$ is $8a + 2^b - 1$. So for the first few odd values of $n$ we have
$$
\begin{array}{c|c}
& \text{maximum number of linearly}\\ 
n &\text{independent vector fields on}\ S^n\\
\hline
1 & 1\\
3 & 3\\
5 & 1\\
7 & 7\\
9 & 1\\
11 & 3\\
13 & 1\\
15 & 8
\end{array}
$$
The above result shows that the answer to the initial question is yes for $k \equiv 0, 1, 3, 7 \bmod 8$.

Consider $M$, the product of the Klein bottle $K^2$ with $k-1$-torus $T^{k−1}$. This manifold is nonorientable, hence, nonparallelizable. On the other hand, it is the total space of a circle bundle over $T^k$, since $K^2$ is a circle bundle over the circle. Let $H$ be a (smooth) connection on this bundle. Take $k$ independent vector fields $X_1,...X_k$ on $T^k$ and lift them to $M$ via the connection $H$. Hence, $M$ admits $k$ independent vector fields. It cannot have $k+1$ vector fields since it is not parallelizable. I am sure there are orientable examples as well, but this requires more work.  
