Final answer:
The compass is the essential tool for constructing a line parallel to another line when only allowed to use a compass and a straight edge, as it can replicate distances and create necessary arcs.
Step-by-step explanation:
The essential tool for constructing a line parallel to line PQ using only a compass and a straight edge is the compass. A ruler can measure distances, and a protractor can find angles, but a compass is the tool traditionally used in geometric constructions for creating arcs and circles necessary for parallel line construction without the need for measurement. When constructing parallel lines, the compasses' ability to replicate distances and create arcs that intersect with existing points or lines is crucial in ensuring the parallelism of the newly constructed line.