I would not split by spaces, there are too many. I would split by "-" and then use regex to extract the date part. Then it's easy with TryParseExact and dd'nd' MMMM yyyy:

string[] textitems = test.Split('-');
string version = textitems[0].Trim();
string edition = textitems[1].Substring(0, textitems[1].IndexOf("(")).Trim();
string dateStr = Regex.Match(textitems[1], @"\(([^)]*)\)").Groups[1].Value;

string[] formats = { "d'st' MMMM yyyy", "d'nd' MMMM yyyy" };
bool validDate = DateTime.TryParseExact(dateStr, formats, CultureInfo.InvariantCulture, DateTimeStyles.None, out DateTime date );

I have added also d'st' MMMM yyyy since i can imagine that this would be your next issue. Another option was to include the brackets in the format: "'('d'nd' MMMM yyyy')'".

You might want to add some code to validate the input first, i have omitted that.

For this I wouldn't even bother with splitting the text, you can do this with a regular expression and named matches.

string test = "Version 21.1.0 - 2021 Edition (22nd August 2021)";
var regex = new Regex(@"Version (?'version'[\d.]+) - (?'edition'\d+) Edition \((?'date'[^)]+)", RegexOptions.None);
var matches = regex.Matches(test);

var version = matches[0].Groups["version"].Value;
var edition = matches[0].Groups["edition"].Value;
var dateString = matches[0].Groups["date"].Value;

// remove date ordinal before parsing
dateString = Regex.Replace(dateString, @"^(\d+)(st|nd|rd|th)", "$1");
var date = DateTime.ParseExact(dateString, "dd MMMM yyyy", CultureInfo.CurrentCulture);

date.ToShortDateString().Dump();

Normally I would use TryParseExact and handle any parse exceptions properly.

You can get an explanation of the main regular expression here: https://regex101.com/r/Nzpa5h/1

I have come up with a solution that combines both approaches. Since the original data is actually a HtmlNode (as indicated at the bottom of the question) and is already split into two span elements, I decided to do it this way:

// The paragraph element should only have two "span" elements
var listSpan = itemParagraph.Descendants("span");
if(listSpan != null)
{
    if(listSpan.Count() == 2)
    {
        // The first "span" element should contain: Version 21.1.0 - 2021 Edition
        var regex = new Regex(@"Version (?'version'[\d.]+) - (?'edition'\d+) Edition", RegexOptions.None);
        var matches = regex.Matches(listSpan.ElementAt(0).InnerText.Trim());

        writer.WriteStartElement("Update");
        writer.WriteAttributeString("Version", matches[0].Groups["version"].Value);
        writer.WriteAttributeString("Edition", matches[0].Groups["edition"].Value);

        // The second "span" element should contain: eg. (22nd March 2021)
        string dateString = listSpan.ElementAt(1).InnerText.Trim(' ', '(', ')');

        string[] formats =
        {
            "d'st' MMMM yyyy",
            "d'nd' MMMM yyyy",
            "d'rd' MMMM yyyy",
            "d'th' MMMM yyyy"
        };

        if (DateTime.TryParseExact(dateString,
            formats, CultureInfo.CurrentUICulture, DateTimeStyles.None, out DateTime dateRevision))
        {
            writer.WriteAttributeString("Date", dateRevision.ToShortDateString());
        }
    }
}

I admit that I do not quite follow how this bit of code actually works:

var regex = new Regex(@"Version (?'version'[\d.]+) - (?'edition'\d+) Edition", RegexOptions.None);
var matches = regex.Matches(listSpan.ElementAt(0).InnerText.Trim());

The above code is modified from one of the supplied answers. But it works. :)

I decided to construct the date using the accepted answer approach as I understand what it is doing, as opposed to the regex suggestion.

@phuzi maybe you could add some explanations or pointers to flesh out your answer concerning the regex syntax?