Default parameter for value must be a compile time constant?

Use regular method overloads instead:

public static void DatesToPeriodConverter(DateTime start, out string date, out string time)
{
    DatesToPeriodConverter(start, DateTime.MinValue, out date, out time);  
}

public static void DatesToPeriodConverter(DateTime start, DateTime end, out string date, out string time) 
{ }

Atlernatively, default(DateTime) is the same as DateTime.MinValue and is compile time constant, but I tend to err away from using this style (there's no guarantee in future that default(DateTime) will equal DateTime.MinValue):

public static void DatesToPeriodConverter(DateTime start, DateTime end = default(DateTime), out string date, out string time)

Or as Marc suggests, use DateTime? which allows a null default value.

DateTime.MinValue is not a const, because the language doesn't like const on DateTime. One option is to use DateTime? instead, i.e.

public static void DatesToPeriodConverter(DateTime start, DateTime? end = null,
     out string date, out string time)
{
    var effectiveEnd = end ?? DateTime.MinValue;
    // ...
}

However, you will still have the issue of having non-default parameters after default parameters - you may need to re-order them to use that as a default.

You can try doing it this way:

public static void DatesToPeriodConverter(DateTime start, DateTime? end , out string date, out string time)
{
    if(!end.HasValue){
        end = DateTime.MinValue;
    }
}

Change a type of the parameter end to a Nullable and use null as a default value:

public static void DatesToPeriodConverter(DateTime start, DateTime? end = null, out string date, out string time)

or use default(DateTime) as a default value:

public static void DatesToPeriodConverter(DateTime start, DateTime end = default(DateTime), out string date, out string time)