Try this, but it seems that there is some difficulty at alpha = Pi: Complex Infinity encountered.

l = 300;

k[\[Alpha]_] := Sin[\[Alpha]/2]^2;
m = 2;
\[Lambda]L = (2*m*EllipticK[k[\[Alpha]]])/l;
x1L[s_] := -s + 
   2/\[Lambda]L (EllipticE[
       JacobiAmplitude[s*\[Lambda]L + EllipticK[k[\[Alpha]]], 
        k[\[Alpha]]], k[\[Alpha]]] - 
      EllipticE[JacobiAmplitude[EllipticK[k[\[Alpha]]], k[\[Alpha]]], 
       k[\[Alpha]]]);
x2L[s__] := -2 k[\[Alpha]]/\[Lambda]L (JacobiCN[
     EllipticK[k[\[Alpha]]] + (s*\[Lambda]L), k[\[Alpha]]]);
Manipulate[
 ParametricPlot[{x1L[s], x2L[s]} /. \[Alpha] -> aa, {s, 0, l}, 
  AxesLabel -> {x, y}, 
  LabelStyle -> {FontSize -> 10, Darker[Black], Bold}, 
  PlotRange -> {-60, 60}], {{aa, Pi/2}, 0, 2 Pi}]

x1L[.2] /. \[Alpha] -> Pi
x2L[.2] /. \[Alpha] -> Pi

Move the definitions of t21, t31, t23, t33 inside the Manipulate, otherwise they will not get updated:

Manipulate[t21 = (2*Pi/lambda[[1]])*l22; t31 = (2*Pi/lambda[[1]])*l33;
  t23 = (2*Pi/lambda[[5]])*l22;
 t33 = (2*Pi/lambda[[5]])*l33; 
 Plot[{2*Y1*Tan[t11] + Y2*Tan[t21] + Y9*Tan[t31], 
   2*Y1*Tan[t13] + Y2*Tan[t23] + Y9*Tan[t33], 
   t11*Y1 + t21*(Y2/2)*(Sec[t21]^2/Sec[t11]^2) + 
    t31*(Y9/2)*(Sec[t31]^2/Sec[t11]^2) - b1, 
   t13*Y1 + t23*(Y2/2)*(Sec[t23]^2/Sec[t13]^2) + 
    t33*(Y9/2)*(Sec[t33]^2/Sec[t13]^2) - b3},
  {Y2, 0.008, 0.05}],
 {Y9, 0.008, 0.05}, {l33, 0.008, 0.012}, {l22, 0.008, 0.012}]

Hi Alexandra,

change the PlotRange to {{-300, 300}, {-100, 100}} , then you see that the right end of the curves moves along the horizontal axis from right to left, the left end stays at the origin.