It is not an input if it does not affect your output (I am sure a better formal definition exists—I usually hear "meaningful", hence my use of that). Without this restriction, any input could be arbitrarily padded, inflating input sizes and making an algorithms complexity appear lower.
I find the even/odd example to be an exceptionally clear case of an algorithm that takes a fixed 1-bit input: Its implementation on a little-endian machine does not know the bounds of its input, only the location of the least significant byte (due to byte-addressing).
Without a defined bound, such an implementation must be assumed to either take a single byte as input (here a byte read is an implementation detail, despite the algorithm only needing one bit), or the full system memory from the location specified as input regardless of contents (due to having used the word "arbitrary", the input is allowed to not fit in registers).
However, I admit that I am now entering deeper theoretical waters with regards to definition details that I am normally comfortable with. I do however not find any other definition to line up with practice.