The man who "cursed like a sailor's parrot" is an unnamed character who was present long ago when Granny realized she had been jilted at her wedding. At that moment, Granny was so overwhelmed that she began to fall in a faint -
The whole bottom dropped out of the world, and there she was blind and sweating with nothing under her feet and the walls falling away. His hand had caught her under the breast, she had not fallen, there was the freshly polished floor with the green rug on it, just as before.
The man who caught Granny, holding her steady until her world comes into focus again, "cursed like a sailor's parrot", and offered to kill the would-be bridegroom for Granny, but Granny begged him not to. The memory of this man, and the whole experience of being left at the altar, comes back to Granny in the waning moments of her life.
The driver of the cart is a little more difficult to identify definitively. By the time Granny speaks of him, she is very near death, and her stream of consciousness is extremely jumbled. The idea of the cart is suggested to her by the sound of Cornelia's voice, which "stagger(s) and bump(s) like a cart in a bad road". From there, Granny envisions herself stepping up in the cart, and finds herself sitting beside a man who is driving. Granny does not look in his face; she knows who he is by his hands, and so she "look(s) instead down the road". By all indications, the man is death, who has come for Granny at last, taking her down the road to her final destination.