Summary: We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for \(GL_n\), as defined by M. Goresky et al. [Represent. Theory 10, 130–146 (2006; Zbl 1133.22013)]. Using a generalization of affine Springer theory for A. Braverman et al.’s Coulomb branch algebras [Adv. Theor. Math. Phys. 22, No. 5, 1071–1147 (2018; Zbl 1479.81043)], we construct a rational Cherednik algebra action on the homology of the Hilbert schemes and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of J. Hilburn et al. [Commun. Math. Phys. 402, No. 1, 765–832 (2023; Zbl 1519.17006)], which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of M. Hogancamp and A. Mellit [“Torus link homology”, Preprint, arXiv:1909.00418], we also show the rational Cherednik algebra acts on the HOMFLY-PT homologies of torus knots. This work was inspired in part by a construction in 3D \(\mathcal{N}=4\) gauge theory.