Consider semi-parametric bivariate copula models in which the family of copula functions is parametrized by a Euclidean parameter θ of interest and in which the two unknown marginal distributios are the (infinite-dimensional) nuisance parameters. The efficient score for θ can be characterized in terms of the solutions of two coupled Sturm-Liouville equations. Where the family of copula functions corresponds to the normal distributios with mean 0, variance 1 and correlation θ, the solution of these equations is given, and we thereby show that the normal scores rank correlation coefficient is asymptotically efficient. We also show that the bivariate normal model with equal variances constitutes the least favourable parametric submodel. Finally, we discuss the interpretation of |θ| in the normal copula model as the maximum (monotone) correlation coefficient.