How to speed up the plotting of B-spline curve?

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.

Re: your error

BSplinePlot2[pts : {{_, _} ..}, knots_, opts : OptionsPattern[{Plot, BSplinePlot}]] := 
 Module[{p = Length@First@Split[knots] - 1, a, b}, 
  {a, b} = {First[knots], Last[knots]};
  With[{a1 = a, b1 = b}, 
   ParametricPlot[
    Total@MapIndexed[NBSpline[First@#2 - 1, p, knots, u] #1 &, pts], {u, a1, b1},
    Evaluate[Sequence @@ FilterRules[{opts}, Options[Plot]]], 
    Epilog -> If[OptionValue[ShowPoints], 
                Join[Text @@@ (Thread@{Table[ Style[Subscript["P", i], 14], {i, 0, Length@pts - 1}], 
                     # + {.4, .2} & /@ pts}), {Red,PointSize[0.015],Point[pts], Green, Line[pts]}]]]]]

pts = {{0, 0}, {0, 2}, {2, 3}, {4, 0}, {6, 3}, {8, 2}, {8, 0}};
knots = {0, 0, 0, 1/5, 2/5, 3/5, 4/5, 1, 1, 1}; 
BSplinePlot2[pts, knots, PlotRange -> {{0, 10}, {0, 5}}, ShowPoints -> True]