James Potter’s parents had a sizable fortune due to Fleamont Potter inventing Sleekeasy’s Hair Potion in 1926. That’s the same potion a young Hermione Granger used to style her hair for the Yule Ball. Fleamont and his wife, Euphemia were elderly by the time they finally had a baby.
"They had quite given up hope of a son or daughter when, to their shock and surprise, Euphemia found that she was pregnant and their beloved boy, James, was born."
James Potter was born 27 March 1960, and his parents lived until 1979 or 1980 at the latest, when they both contracted dragon pox and died, leaving their wealth to a young James, who married Lily Evans when he was just eighteen years old. Some wonder if they married because a baby was on the way, but I think James wanted a wife to share his life and not feel so alone. He didn’t have siblings.
They were married when they were eighteen and Lily Evans Potter did not become pregnant with Harry until the autumn of 1979 when the couple was nineteen. Seems young, but back then, some people married right after high school. Now one’s adolescence extends into a person’s mid-twenties or early thirties, so it seems strange that two teens without families would marry and have a baby, especially since a War against Voldemort was being waged. But during wars, many people do not put off marriage. They rush into it.
As to Lily Evans’ parents, not much is known. Mr. Evans and his wife (no name is known for her) had Petunia in the late 1950s. The parents and Petunia were Muggles, but they gave birth to a talented witch, Lily Evans on 30 January 1960. We know Mr. and Mrs. Evans were very pleased to discover their daughter, Lily, was a witch and would be attending Hogwarts. On 1 September 1971, Lily’s parents took her to Platform 9 3/4 to the Hogwarts Express. They were so happy for her and them.
Sometime after that, Mr. Evans died a Muggle death, which could be cancer or a heart attack, no one knows, between 1971 and 1979. Lily’s mother also died a Muggle death sometime before her own daughter was murdered by Voldemort. Not much is known about Lily and Petunia’s mum, but since an orphaned Harry had to live with Lily’s sister, the only family baby Harry had left, we know Lily’s mother had died before 31 October 1981.
Muggles do not have the long lifespan of many wizards and witches, so the younger Evans dying shortly before the elderly Potters died makes sense. It’s sad but not unheard of for teens to be “orphaned” so young.
I wanted to add that Lily’s mum and dad spoke of her so often and were mesmerized that they had a witch in the family, that it made Petunia jealous and sad and she felt less wanted and undesired. This does not in any way justify her treatment of Harry, her nephew, while he lived with them, but it does explain her actions a bit more. She’s human. Petunia made a mistake, but her parents were not alive to help her. They would have been great raising Harry Potter, but it never happened.