Log your results to a file (or use amx_log) after the 'parse' native and then look what's wrong and why your strings aren't equal.
Also, some stuff:
1. Use the new file system instead of the old one, it's much better.
2. Move the get_user_authid before the for loop starts, because now your basically calling this native each 'line' and it's the steamid never changes.
3. Remove the get_user_authid inside the 'else if' part
4. You can write 'else' instead of 'else if (!equal..', the conditions are the same.
5. When writing strings to a file (such as SteamIDs), use quotes ("%s"). it will prevent some bugs with the parse function and spaces inside the string.