It is well known that likelihood inference for arbitrary nonlinear diffusion processes observed at discrete times is problematic since closed form transition densities are rarely tractable. One widely used solution involves the introduction of latent data points between every pair of observations to allow a sufficiently accurate Euler-Maruyama approximation of the true transition densities. In recent literature, Markov chain Monte Carlo (MCMC) methods have been used to sample the posterior distribution of latent data and model parameters; however, naive schemes suffer from a mixing problem that worsens with the degree of augmentation. We will consider some recently developed MCMC schemes that are not adversely affected by the amount of augmentation. In particular, by sampling parameters conditional on a skeleton of the driving Brownian motion rather than the sample path, the mixing problem can be overcome. The methodology will be illustrated by estimating parameters governing the diffusion approximations of some interesting systems biological models.