“His parents will never believe but what he was enticed away by his uncle.” Omit what. The use of but that would be equally objectionable. But is sufficient.
A reconstruction of the sentence would improve it. “His parents will always believe,” or “Will never cease to believe that,” etc.
Doubt not but
“I doubt not but your friend will return.” Say, “I doubt not that your friend will return.”
Not impossible but
“It is not impossible but he may call to-day.” Use that instead of but.
Whether, Whether
“Ginevra has not decided whether she will study history or whether she will study philosophy.” As there is nothing gained in clearness or in emphasis by the repetition of “whether she will,” this shorter sentence would be better: “Ginevra has not decided whether she will study history or philosophy.”
As though