So there was a little confusion here in the responses due to the fact that there are few different modes of gameplay (regular play, inspecting, fort scythe, and Mystic) but Devon's last response here is correct.
I consider that there are two separate "modes" of being a scythe in Eyewire. Reaping/reap-growing and scythe-completing. They should be done separate from each other. This is the case especially in forts. As you stated, "you cannot complete what you reap, it's like a student correcting himself, especially right after reaping". Ultimately, Eyewire is a community based game with multiple people building a consensus together from their individual solutions in each cube. This is true in regular play as well as scything/reap-growing/scythe-completing.
Mode 1: Reapgrowing should be declared with where you are working in a fort in chat.
Mode 2: Scythe-completing should also be declared in any cell in chat. This is where you would use the suggestion of "completing parents periodically so that other players see a purple trail that belongs to you".
I hope this helps makes things more clear in addition to the responses by Nseraf and Atani.