A lot depends on your soil, Chantenay. If you have very heavy, sticky soil, they are best lifted and stored - for preference I would box them in peat - in temperature conditions suitable for potatoes.
My soil is mainly light to medium, and I leave both Carrots and Parsnips in the ground until needed. A covering of horticultural mesh over the carrots will protect against surface predators, and will also make them much easier to lift when the ground is frozen or snowbound.
If you feel the carrots are mature then you can certainly use coffee on them straight away, although I usually wait until early September. However, I don`t advocate coffee for parsnips for 2 reasons:-
1. I`ve never yet had parsnips attacked by slugs - any damage mine get is usually from carrot fly or millipedes,
2. Parsnips will continue to grow until November; coffee might well inhibit this.