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In the Harry Potter series, Harry deals with death in a multitude of ways, consistently adjusting his understanding of it as he experiences extraordinary situations. From the start, he's an orphan whose mother gave up her life for him, branding him with a protection spell that kept him safe from Voldemort's death curse as a baby. His mother's love continued to keep him safe even when Voldemort returned in the movie Harry Potter and the Goblet of Fire, and Harry's parents even returned in a magical, but mysterious way.
During this reunited scene between Voldemort and 14-year-old Harry, the two duel for Harry's life, and ghosts seem to appear from the connection of their wands. Harry's parents, Lily and James, were some of these spirits, and they spoke to him in this form, urging him to run while they rushed Voldemort long enough for him to escape. They saved his life, but the event opened a lot of questions for viewers. The most important questions are how his parents appeared and if he could access them at will.
Where Harry Potter Saw His Parents Before
Lily and James had returned to him previously in the series, appearing in the Mirror of Erised in Harry Potter and the Philosopher's Stone while he was exploring the castle with his invisibility cloak. Dumbledore found Harry while the boy stared at his parents in the mirror, and Dumbledore advised him not to return again, making it clear that focusing on the dead does not benefit the present. The mirror can only show someone what they wish to see, and many wizards had gone mad while staring into the mirror, not wanting to leave behind their most potent desires.
Harry had asked Hagrid about bringing his parents back, but Hagrid made it clear to Harry that not all things can be solved with magic. Working with the dead could have dark consequences, so it's better to release the attachment and move forward. Harry was disappointed by this, especially when he thought he saw his father in Harry Potter and the Prisoner of Azkaban before realizing he was really just seeing a future version of himself.
How Harry Potter's Parents Made an Appearance in Goblet of Fire
Through this acceptance, Harry stopped expecting his parents to appear, but they did arrive one more time in The Goblet of Fire from the connection between Voldemort and Harry's wands. The apparitions of Lily and James weren't actually ghosts or spirits, but they were considered "echoes" produced by Voldemort's wand. This is called a Reverse Spell (Priori Incantatem), which forces a wand to show an "echo" of the last time the spell was performed. Since Voldemort had used his wand to kill Harry's parents and Cedric Diggory, they appeared as echoes from the wand while he was losing the duel against Harry.
Harry and Voldemort share the same core of their wands, binding them in a new way from the various other events that have bonded them together. Harry might have been young, but he was a powerful wizard who scared Voldemort. Their duel took place in a graveyard, and Voldemort cast the killing curse before Harry blocked it. Every time Voldemort cast that same spell was retained by the wand and provided the path for the echo to travel through.
What differentiates echos from ghosts is the fact that echos retain the "appearance and character" of the deceased person, as well as memories from their last moments before death. The echoes can behave according to that knowledge, but they are not the actual deceased person's spirit. Echoes also can only linger on the mortal plane for a few moments before they fade back to the beyond, so that's why his parents pushed him to run while they used the last of their energy to protect him. For these reasons, the moment was a phenomenal one but not something that could ever be replicated. It was a once-in-a-lifetime experience for Harry to interact with the echoes of his parents right before they saved his life.