  Minimal cubature rules of degree $4n-1$ for the weight functions $$
W_{\a,\b,\pm \frac12}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} ((1-x^2)(1-y^2))^{\pm
\frac12} $$ on $[-1,1]^2$ are constructed explicitly and are shown to be closed
related to the Gaussian cubature rules in a domain bounded by two lines and a
parabola. Lagrange interpolation polynomials on the nodes of these cubature
rules are constructed and their Lebesgue constants are determined.
