How is Vertex Cover reducable to Independent Set using parametrized reduction with parameter k?
It seems your confusion comes from the fact that Lemma 5.5 is a bit of a strange statement. In fact, we can replace Lemma 5.5 with the statement that for any problem $B$ such that $B_Y$ is a yes-instance of $B$ and $B_N$ is a no-instance of $B$ we have that $A\leq_{\mathrm{fpt}}B$. The proof of this adapted lemma is precisely the same as Lemma 5.5, only now we use $B_Y$ in the second bullet-point and $B_N$ is the third.

The crucial reason why this works is that we already know that $A$ is fixed parameter tractable. In a sense, this lemma shows that we learn nothing from a fpt-reduction from a problem we already know is fpt. 

So, this lemma is in fact useless when trying to create a reduction from a problem we don't know is fpt, such as $k$-Clique. 

When you try to apply Lemma 5.5 when $A$ is not fpt (or at least not known to be), you fail to satisfy part 3 of the definition of an fpt-reduction.