  We revisit classic string problems considered in the area of parameterized
complexity, and study them through the lens of dynamic data structures. That
is, instead of asking for a static algorithm that solves the given instance
efficiently, our goal is to design a data structure that efficiently maintains
a solution, or reports a lack thereof, upon updates in the instance.
  We first consider the Closest String problem, for which we design randomized
dynamic data structures with amortized update times $d^{\mathcal{O}(d)}$ and
$|\Sigma|^{\mathcal{O}(d)}$, respectively, where $\Sigma$ is the alphabet and
$d$ is the assumed bound on the maximum distance. These are obtained by
combining known static approaches to Closest String with color-coding.
  Next, we note that from a result of Frandsen et al.~[J. ACM'97] one can
easily infer a meta-theorem that provides dynamic data structures for
parameterized string problems with worst-case update time of the form
$\mathcal{O}(\log \log n)$, where $k$ is the parameter in question and $n$ is
the length of the string. We showcase the utility of this meta-theorem by
giving such data structures for problems Disjoint Factors and Edit Distance. We
also give explicit data structures for these problems, with worst-case update
times $\mathcal{O}(k2^{k}\log \log n)$ and $\mathcal{O}(k^2\log \log n)$,
respectively. Finally, we discuss how a lower bound methodology introduced by
Amarilli et al.~[ICALP'21] can be used to show that obtaining update time
$\mathcal{O}(f(k))$ for Disjoint Factors and Edit Distance is unlikely already
for a constant value of the parameter $k$.
