Representing and Parsing an Open or Closed Range
I would check the result of range


range = _RemoveSpaces.Replace(range, string.Empty);



against string.Empty because that is one of the cases where the code reaches this. 


// No idea what we were given, give a completely open range.
return new Range&lt;T&gt;(null, null);  



Returning early will get you some performance here. Doing a null check on range before that call isn't necessary because Regex.Replace() throws a ArgumentNullException for the case that range == null.  

Speaking about Regex.Replace I would prefer string.Replace() over Regex.Replace because it seems faster. See: comparing regex replace string replace and stringbuilder replace which has better performance



One should not always do what can be done.


public override string ToString()
{
    return (this._Minimum == null) &amp;&amp; (this._Maximum == null)
        ? string.Empty
        : (this._Minimum == null
            ? string.Format("-{0}", this._Maximum)
            : (this._Maximum == null
                ? string.Format("{0}+", this._Minimum)
                : string.Format("{0} - {1}", this._Minimum, this._Maximum)));
}  



IMO this isn't readable at first glance so maybe some good old if..else would be better. Having underscore prefixed membervariables and using this is too much.  

public override string ToString()
{
    if ((_Minimum == null) &amp;&amp; (_Maximum == null)) { return string.Empty; }

    if (_Minimum == null) { return string.Format("-{0}", _Maximum); }

    if (_Maximum == null) { return string.Format("{0}+", _Minimum); }

    return string.Format("{0} - {1}", _Minimum, _Maximum)));
}  


If you are using C# 6 you could use the string-interpolation operator $ instead of string.Format().  



About Interval vs Range I would prefer Range because Interval IMO suits only with dates and times but e.g don't suits with int.  

__  

Inside the Parse() method I would check the found minimum and maximum before calling the constructor. IMO throwing an exception in a constructor should be avoided if possible.  



In general you code looks tidy and clean. 


You are using good variables- and methodnames. 
You are using braces although they are optional. 
Your code is easy to read.