Robert Charles Anderson (the definitive source for PGM and possibly the GOAT of genealogists) wrote "The English Origin of John Hunting (1602-1689) of Dedham, Massachusetts" for the National Genealogical Society Quarterly v78 in June 1990.
In it, he points out that the 1567 marriage record for Roger Hunting with "Jania [Jane] Leche" is in Latin and no one is certain what surname is even meant by it. The closest he comes to identifying a family of origin for her is to point out that a Leach/Leech family existed in the vicinity. He does not name her parents.
Jane's profile on WikiTree (don't blame me, I just adopted it!) names an "Unknown (Massey) Leche" as her mother, who just happens to be descended from the Venables and Stanleys and so on back to Magna Carta surety baron William Malet, and William Leche as her father, who just happens to be descended from the Mainwarings and Venables and so on back to Magna Carta surety baron Robert FitzWalter.
I'm sure you'll be as shocked as I am that no evidence is provided in support of these claims.
Before I rashly disconnect them into the ether, is there any more recent research that would validate these or any other parents of Jane (Leche) Hunting?