Given a graph $G$, the hard-core model defines a probability distribution over its independent sets, assigning to each set of size $k$ a probability of $\frac{λ^k}{Z}$, where $λ>0$ is a parameter known as the \emph{fugacity} and $Z$ is a normalization constant. The Glauber dynamics is a simple Markov chain that converges to this distribution and enables efficient sampling. Its \emph{mixing time}, the number of steps needed to approach the stationary distribution, has been widely studied across various graph classes, with most previous work emphasizing the dichotomy between polynomial and exponential mixing times, with a particular focus on sparse classes of graphs. Inspired by the modern fine-grained approach to computational complexity, we investigate subexponential mixing times of the Glauber dynamics on geometric intersection graphs, such as disk graphs. We further study parameterized mixing times with respect to two structural parameters that can remain small even in dense graphs: the tree-independence number and the path-independence number. Building on a result of Dyer, Greenhill, and Müller, we show that Glauber dynamics mixes in polynomial time on graphs of bounded path-independence number, and in quasi-polynomial time when the tree-independence number is bounded. Moreover, we prove that both bounds are tight via a conductance argument, thereby resolving a question raised in their work. This work provides a simple and efficient algorithm for sampling from the hard-core model. Unlike classical approaches that rely explicitly on geometric representations or on constructing decompositions such as tree decompositions or separator trees, our analysis only requires their existence to establish mixing time bounds-these structures are not used directly by the algorithm itself.