A parameter is one of the variables that gets assigned a coefficient in the statistical model. For example, a quantitative variable in a regression or a logit model typically gets one coefficient, or parameter in the model, while a categorical variable with k categories will get k-1 coefficients/parameters. (because the k levels will be represented as k-1 dummy or effects coded variables).
It's not so much the number of parameters that matters in this case as the correlation patterns among them, which tend to be better with alternative-specific designs than with designs built with prohibitions.
