It's entirely possible that the War Machine suit DID have a parachute mechanism, but it was likely an automatic or AI-controlled one. In the specific case of War Machine's accident, the suit's power source was destroyed - both the suit and the AI were completely dead. Rhodey was a falling tank at that point, so even a traditional parachute wouldn't have helped him... he couldn't move his arm to pull the ripcord.
Or, as an alternate explanation, one could debate that the Iron Spider suit had a parachute directly BECAUSE of what happened to Rhodey in "Civil War". Either idea works.