Well, the code that "does" it is functionally identical in the storefront and the admin, and it's nothing extra special in the code:
Code:
if (STORE_PAGE_PARSE_TIME == 'true') {
error_log(strftime(STORE_PARSE_DATE_TIME_FORMAT) . ' - ' . $_SERVER['REQUEST_URI'] . ' (' . $parse_time . 's)' . "\n", 3, STORE_PAGE_PARSE_TIME_LOG);
}
So, if one isn't doing it, then it's probably not getting called, due to an override or something. Most often any failure with logging is due to a permissions problem, but if the admin can write fine then permissions is not likely the issue. Thus it suggests that you have an override of the footer module that doesn't have that code in it.
What do you have configured for the Parse Time Log filename in your admin? Is it an absolute path name or is it a relative path, meaning the logging is being done to different destinations based on admin vs storefront?