Three parameters with 20 values each means 8,000 backtests. Five parameters means 3.2 million. The compute is only half the problem; the other half is that you have now run 3.2 million trials and the best one will look spectacular through pure multiple-testing.
Grid search is nonetheless the clearest way to visualise parameter-sensitivity with one or two parameters, because you can plot the surface and see whether there is a parameter-plateau.
Keep grids coarse. Testing lookbacks of 10, 20, 40, 80, 160 tells you almost everything that testing 10 through 200 in steps of 1 would, at a fortieth of the trials, and the coarse grid is much harder to fool yourself with.
Related: parameter-optimisation, random-search, bayesian-optimisation, multiple-testing