Origins
A parade is, etymologically, a place where people stop. The word arrived in English from French parade in the mid-17th century, but its deeper history lies in Spanish parada — 'a halt, a stopping place' — from parar, 'to stop, to prepare', from Latin parāre, 'to prepare, to furnish'.
The military meaning came first. A parade ground was where troops halted and assembled for inspection. Officers would review their soldiers standing in formation — no movement, no music, no floats. The spectacle was discipline, not celebration.
Development
The shift to 'festive procession' happened as the display aspect overtook the military one. If the point of a parade was to show off your forces, why not show off your community? By the 18th century, parades had become public events marking holidays, victories, and civic pride.
Latin parāre is one of the most productive roots in English. Prepare means 'to make ready before'. Repair means 'to make ready again'. Separate means 'to prepare apart'. Apparatus is equipment prepared for a task. Even parry in fencing descends from the same root — to parry is to prepare a defensive position against a blade.