What parallels does this story share with "The Rime of the Ancient Mariner" so far?

I am not sure where you are in Frankenstein, but both that novel and "The Rime of the Ancient Mariner" use a frame narrative. In other words, in both cases, the title characters tell their stories to a person they have never seen before. In the case of the Mariner, he relates his tale to a wedding guest. The guest is distressed that he can't get to the wedding festivities, but is nevertheless transfixed by the strange Mariner and compelled to listen to his story. In Frankenstein, Captain Robert Walton finds Victor Frankenstein half starved and floating on an ice floe in the Arctic. He too listens to a very strange saga, which he writes about in letters to his sister.
In both cases, too, the stories are about the suffering that is caused by each man—Victor and the Mariner—acting out of pride or arrogance. The Mariner violates the laws of God and man and brings down a curse when he arrogantly kills the albatross that saved his ship. Frankenstein, out of a sense of pride, violates God's law when he creates life out of inanimate body parts. He compounds the problem when he rejects his creation, who out of anguish at being unloved, kills the people closest to Victor. Both the Mariner and Victor live to regret what they have done.