For an integer $ρ$ such that $1 \leq ρ\leq v/3$, define $β(ρ,v)$ to be the maximum number of blocks in any partial Steiner triple system on $v$ points in which the maximum partial parallel class has size $ρ$. We obtain lower bounds on $β(ρ,v)$ by giving explicit constructions, and upper bounds on $β(ρ,v)$ result from counting arguments. We show that $β(ρ,v) \in Θ(v)$ if $ρ$ is a constant, and $β(ρ,v) \in Θ(v^2)$ if $ρ= v/c$, where $c$ is a constant. When $ρ$ is a constant, our upper and lower bounds on $β(ρ,v)$ differ by a constant that depends on $ρ$. Finally, we apply our results on $β(ρ,v)$ to obtain infinite classes of sequenceable partial Steiner triple systems.