How can I parse a String and return an Enum value in Apex Code?

This isn't generic, but it would work:

String dayOfWeekNameToMatch = 'Wednesday';
DayOfWeek dayOfWeekMatch;
for (DayOfWeek dow: DayOfWeek.values()) {
    if (dow.name() == dayOfWeekNameToMatch) {
        dayOfWeekMatch = dow;
        break;
    }
}