Let $C$ be a smooth, projective, geometrically integral hyperelliptic curve of genus $g \geq 2$ over a number field $k$. To study the distribution of degree $d$ points on $C$, we introduce the notion of $\mathbb{P}^1$- and AV-parameterized points, which arise from natural geometric constructions. These provide a framework for classifying density degree sets, an important invariant of a curve that records the degrees $d$ for which the set of degree $d$ points on $C$ is Zariski dense. Zariski density has two geometric sources: If $C$ is a degree $d$ cover of $\mathbb{P}^1$ or an elliptic curve $E$ of positive rank, then pulling back rational points on $\mathbb{P}^1$ or $E$ give an infinite family of degree $d$ points on $C$. Building on this perspective, we give a classification of the possible density degree sets of hyperelliptic curves.