A Language Belonges to PSPACE
No. 
Note that given languages $A,B$ and a reduction $A\le_L B$, you get that $B$ is at least as hard as $A$, but that's about it.

You can contradict the above statement with the following extreme:

Take $A = \{0\}$, then $A\in \text{PSPACE}$, and take $B= H_{TM}\cup\{0\}$.

Define $f$ to be the following reduction $A \le_L H_{TM}\cup\{0\}$ (we can assume that $1\notin H_{TM}$):

$$f(0) = 0$$ $$\forall x\in \Sigma^*\setminus\{0\}  . f(x)=1$$.

Clearly $f$ works in logarithmic space, but $B$ is clearly not in $\text{PSPACE}$. In fact, it isn't even in $\text{R}$.