Reduction between parametrized problems Can we construct reduction from $k$-sum to $l$-clique or vice versa where $k$ and $l$ are some fixed integers?

That is given two parametrized problems whose unparametrized version is $NP$-complete when can we construct reductions between them?

In general NP complete problems the reductions are polynomial time.

What is the corresponding analogue in parametrized problems?
You are asking two questions, I will answer the second. Parameterized reduction from problem $P$ to $Q$ is a function $\phi$ with the following properties : 


$\phi(x)$ can be computed in time  $f (k)·|x|^{O(1)}$, where $k$ is the parameter of $x$,
$\phi(x)$ is a yes-instance of $Q$ if and only if $x$ is a yes-instance of $P$.
If $k$ is the parameter of $x$ and $k′$ is the parameter of $\phi(x)$, then $k′ ≤ g(k)$ for some function $g$.



  Note that $f(k)$ need not to be a polynomial function in $k$, it could be like $2^k$ or $c^k$, where $c$ is constant.


Example: INDEPENDENT-SET instance $(G,K)$ into CLIQUE instance $(G',K)$ is a parameterized reduction. You can verify all three conditions mentioned above.

Reference : http://www.cs.bme.hu/~dmarx/papers/marx-warsaw-fpt3