  For an integer $\rho$ such that $1 \leq \rho \leq v/3$, define
$\beta(\rho,v)$ 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 $\rho$. We obtain lower bounds on $\beta(\rho,v)$ by giving explicit
constructions, and upper bounds on $\beta(\rho,v)$ result from counting
arguments. We show that $\beta(\rho,v) \in \Theta (v)$ if $\rho$ is a constant,
and $\beta(\rho,v) \in \Theta (v^2)$ if $\rho = v/c$, where $c$ is a constant.
When $\rho$ is a constant, our upper and lower bounds on $\beta(\rho,v)$ differ
by a constant that depends on $\rho$. Finally, we apply our results on
$\beta(\rho,v)$ to obtain infinite classes of sequenceable partial Steiner
triple systems.
