The plot is sped up substantially if you use Evaluate:
ParametricPlot[
Evaluate[ Total@MapIndexed[NBSpline[First@#2 - 1, p, knots, u] #1 &, pts]] ,
{u, a, b}, opts]
(I only looked a trial 1 , but I think your other try have the same issue )
It helps a little more if you remove the Simplify from NBSpline and simplify the whole thing:
ParametricPlot[
Evaluate[Total@
MapIndexed[NBSpline[First@#2 - 1, p, knots, u] #1 &, pts] //
Simplify], {u, a, b}, opts]
Your original form is a sum of piecewise expressions. The outer Simplify condenses the whole thing into a single piecewise.
The gaps seem to relate the nature of the discontinuity in derivatives of the bspline w/ respect to its parameter at the knots, which evidently fools Parametric Plot into thinking there is an actual discontinuity.
The gaps close with PlotPoints -> 1000 , though if you look at the graphics produced you'll see you still have separate Lines for each portion. I don't think there is anything to do about that except not use ParametricPlot.
You might try doing away with ParametricPlot and doing Graphics@Line@Table .. ,
which may speed it up as well.