In the parasite problem, a particle ('ant') diffuses randomly on a random percolation cluster in the limit of concentration to 0 ('lattice animal'). Monte Carlo simulations and scaling arguments show that for large animals the distance r travelled by this parasite increases as t(1/zA) with time t. The authors find zA approximately=3.4 on the simple cubic lattice and zA approximately=2.6 on the square lattice. This anomalous diffusion is in rough agreement with a generalisation of a suggestion by Alexander and Orbach (1982). Heuristic arguments in favour of this suggestion are given. Also, they look at corrections to scaling for concentrations equal to the percolation threshold.