Two subspaces $A$, $B$ are paralel, if and only if they are both subspaces of a space with a dimension equal $d = 1 + \max (\dim A, \dim B)$? It is true, that two affine subspaces $A$, $B$ are paralel, if and only if there exists an affine space $C$, that both are subspaces of $C$, and $\dim C = 1 + \max (\dim A, \dim B)$?
If so, do You have a proof?
 A: I guess, you mean affine subspaces. 
Well, the statement is not true: consider for example any two lines as $A$ and $B$ intersecting in a point, then they satisfy your criterium, as they generate the plane of those intersecting lines, though the lines are not parallel.
Another example is $A:=$ plane $(z=0)$, $B:=$ $z$-axis.
However, if in addition $A\cap B=\emptyset$ is assumed, then we can conclude that $A\parallel B$, as follows:
Without loss of generality, we can assume that $\dim A\ge\dim B$ and that $0\in A$ (else we can shift both sets with $-a_0$ for any fixed $a_0\in A$). Then, let $b_0\in B$ arbitrary. By assumption, $b_0\notin A$, so for the generated (now linear) subspace $W:=\langle A,b_0\rangle=\{a+rb_0\,\mid\, r\in\Bbb R,\ a\in A\}$, we have
$$\dim W = 1+\dim A=1+\max(\dim A,\dim B)$$
So, that $W=\langle A,B\rangle$ by hypothesis. This means that for any $b\in B$, we have $b=a+rb_0$. I let it for you to check that if $r\ne 1$, then the line of $b_0$ and $b$ will intersect $A$, which case is excluded.
So, we have $r=1$ and $b-b_0\in A$, meaning that the vector from $b_0$ to $b$ is parallel to $A$. -QED-
