Preheat the oven to 350 degrees.
Add the coconut flour, oats, brown sugar, and parsley to a medium bowl.
Peel and mash the ripe banana with a fork or potato masher. Add this to the bowl.
Wash and peel the two carrots and shred them with a box or plane grater. Add this to the bowl.
Melt the coconut oil in the microwave and add this to the bowl. (You can also add it without melting, which is what I did.)
Crack an egg into the bowl.
Add the water as you are mixing all of the ingredients together with a spoon. (If too dry, add a little more water, but if too sticky, add a little more coconut flour. You can also try adding a little rice or almond flour.)
Knead the dough with your hands. You want a play-doh-like consistency.
Form into a dough ball easy enough to work with. You may choose to work with one big dough ball or several smaller ones.
Flour the top of a counter space and roll the dough to about an ⅛ inch thick.
Cut the dough using cookie cutters of your choice or form into round little discs.
Bake for 10 to 12 minutes. (Ovens vary, so start checking them at 9 minutes.)
Let the cookies thoroughly cool before serving to your dog.