Has parameterized complexity led to better algorithms?
There are several examples of problems where a parameterized algorithm performs well in practice. Let me mention two such problems.
In the $k$-Path problem where we are looking for a simple path of length $k$. Alon, Yuster and Zwick [1] showed that this problem can be solved in $2^{O(k)}\cdot n$ time on $n$-vertex graphs. A weighted version of $k$-Path has applications in computational biology and the biologically interesting paths have length at most 20. Several of the implemented algorithms successfully use the color coding technique [2].
The Clique problem parameterized by the solution size $k$ has presumably no FPT algorithm, but Clique parameterized by the degeneracy $d$ of the input graph has an FPT algorithm. More precisely, all maximal cliques of an $n$-vertex graph can be enumerated in $O(3^{d/3}\cdot n)$ time [3]. Since many real-world graphs (e.g. social networks) have small degeneracy, this running time bound explains why clique enumeration is feasible on these graphs.
I would say there is a crucial difference between the two results. In the case of $k$-Path, the color coding technique is an FPT technique that was developed in theory and was later turned into a practical algorithm. In the case of Clique, the central technique of the FPT algorithm, which, roughly speaking, is to enumerate first the cliques containing a minimum-degree vertex $v$ and then enumerating all cliques not containing $v$, was already known, in some form, before the theoretical analysis and probably used in several implementations. Hence, I would say that for $k$-Path, parameterized algorithmics has led to better algorithms, and for Clique, parameterized algorithmics rather explains why algorithms are good.
[1]: Alon, Yuster and Zwick: Color Coding.  J. ACM 42(4): 844-856 (1995) https://doi.org/10.1145/210332.210337
[2] Jacob Scott, Trey Ideker, Richard M. Karp, Roded Sharan: Efficient Algorithms for Detecting Signaling Pathways in Protein Interaction Networks. J. Comput. Biol. 13(2): 133-144 (2006)
[3] David Eppstein, Maarten Löffler, Darren Strash:
Listing All Maximal Cliques in Large Sparse Real-World Graphs. ACM J. Exp. Algorithmics 18 (2013)