Drilling machines are named by their construction and the way spindles are arranged. The question asks specifically for a machine built with several spindles so that different operations run at the same time, which points to a multi-spindle layout.

A gang drilling machine has a number of independent spindle heads mounted side by side over one long common table, each head set up for a different operation.

Typically the first spindle drills, the next counterbores, the next reams and a further one taps; the work is slid along the table from one spindle to the next so successive operations are completed in sequence on the same setup.

This side-by-side, multi-spindle arrangement is exactly what lets different drilling operations proceed together, matching the description.

A bench drilling machine is a small single-spindle unit for light work, and an upright drilling machine is a larger but still single-spindle floor machine, so neither offers multiple simultaneous operations.

A radial drilling machine also has one spindle, carried on an arm that swings and moves over large work; its flexibility is in reach, not in running several spindles at once.

Hence, the answer is Gang drilling machine.