Usage of the phrase "if will"

[Ofriendragon]

--- How about going to the cinema with me tonight?
---I’ll ask my parents if they ____ me go.
A. have let
B. let
C. will let
D. are going to let
The answer is C

Is the will a must? Can B be possible?
I have been taught that I should use "if + simple present" and I should not use "if + future tense" except the case I use "will" to mean "willingness". What does the will mean here?

Many thanks in advance.

[Mister Micawber]

.
I think it does indeed mean willingness. You might note, however, that the 'll form is more common: if they'll let me go.
.
_________________
Native English teacher at Mister Micawber's

[Ofriendragon]

Thank you very much, Mister Micawber.