In Ralf Schmidt's appendix to "Jacquet-Langlands-Shimizu correspondence for theta lifts to $\mathrm{GSp}(2)$ and its inner forms" by Narita and Okazaki , he computes the representations of $\mathrm{GSp}(1,1)$ by examining the tables in his book and discarding the representations whose Weil-Deligne representations fall in the Siegel parabolic. He obtains the answer of IIa, IVa, IVc, Va, Vb, Vc, and VIc.

When I try to replicate the calculation I run into the following problem: representations of type VIa and VIb look almost identical to those of Va. But Va is included and VIa and VIb are not. Furthermore, as written almost all the nilpotent matrices seem to fall neatly into the Siegel parabolic, and the representations are diagonal.

My guess is that we have to compute the image of the Galois representation, and that this cannot be easily read off from the Weil-Deligne representation. Unfortunately I've not succeeded in seeing how to do this either: the discussion in chapter $2$ of Local Newforms for $\mathrm{GSp}(4)$ is not helping.