A "part" is any type of measurement, such as ounces, jiggers or cups. A part is a ratio cue that allows bartenders to scale recipes easily to make multiple drinks or large batches instead of just one cocktail.
When mixing a single drink, one part is usually the equivalent of 1 ounce. A classic vodka martini, for example, uses six parts of vodka and one part of dry vermouth, which is the equivalent of 6 ounces of vodka and 1 ounce of dry vermouth. To make a pitcher of martinis using 1/2-cup measurements as the parts instead of ounces is the same as mixing 3 cups of vodka and 1/2 cup of dry vermouth.