For an integer ρ such that 1≤ρ≤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)∈Θ(v) if ρ is a constant, and β(ρ,v)∈Θ(v2) 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.