One of the big challenges could be Harry's own internal struggle. He has to balance his longing for his parents with the knowledge that what he's trying to do might go against the natural order of things. He might also have to deal with the guilt of potentially using dark magic or going against the rules. And then there's the matter of his friends. Some of them might support him, but others might be afraid of the consequences and try to dissuade him, which would put a strain on his relationships.
He might face opposition from the wizarding authorities. Resurrecting the dead is generally not allowed in the wizarding world, so he could get into a lot of trouble.
They could use magic spells that are long forgotten. Maybe they find a way to communicate with Harry through his dreams, guiding him to a place where they can be reunited.
In some fanfictions, magic might be used to bring them back. For example, a powerful time - turning spell could be involved, allowing them to be retrieved from the past before their deaths. Another way could be through some sort of deal with dark forces, although this would likely have many negative consequences.
One common plotline is the search for a way to reverse the death spell. Harry and his friends might go on a quest to find rare ingredients or spells. Another is that there could be a prophecy about Harry being the one to bring his parents back, which drives the story forward. Also, there could be conflicts with the Ministry of Magic, as they might oppose any attempts to bring back the dead.
Snape might face the challenge of his own biases. He has had a long - standing dislike for Harry's father, and this could initially cloud his ability to help Harry. Also, the wizarding community may not fully understand autism, so Snape could face resistance or lack of support when trying to implement special accommodations for Harry.
One challenge could be not being able to change things as easily as he thinks. Time might have its own rules. Another is that he might accidentally change something that has a much worse consequence later.
One challenge could be the Time - Turner rules. If they are using a Time - Turner, they need to be very careful not to break any of the laws regarding time travel. For example, they can't be seen by their past selves or make significant changes that would disrupt the future they come from. Another challenge is the danger of the time period they visit. If they go to a time when Voldemort was active, they are at great risk of being detected by Death Eaters.
One challenge would be the goblins. They are very protective of Gringotts and would not easily give up control. They have their own laws and traditions regarding ownership. Harry would have to deal with their resistance and find a way to convince them or negotiate with them.
One challenge would be dealing with Snape's harsh criticism. Snape is not known for being gentle with his words, and Harry would have to learn to take it constructively. Another challenge could be the secrecy involved. Snape has a lot of secrets, and Harry would need to respect that while also trying to figure out what Snape is really thinking. Also, other students might be jealous or suspicious of Harry's new relationship with Snape, which could create some social problems for Harry at Hogwarts.
One challenge could be cultural differences. Harry is from a world where magic is more about spells and potions, while Zatanna's magic is more about incantations and showmanship. Another challenge might be the acceptance of their relationship by their respective communities. Some in the wizarding world might think Zatanna's magic is too different and unorthodox, and some in Zatanna's world might not understand the ways of the wizarding world. Also, they might face villains who target them because of their combined power and the threat they pose to the status quo.
Another challenge for Harry as Dada teacher could be dealing with the school's bureaucracy. He might want to introduce new teaching methods or change the curriculum, but he has to go through the proper channels. And there could be resistance from some of the more conservative teachers or the school administration who are used to the old ways of teaching Dada.