Install-Module -Name PSParseHTML -AllowClobber -Force
Force and AllowClobber aren't necessary, but they do skip errors in case some appear.
Update-Module -Name PSParseHTML
That's it. Whenever there's a new version you simply run the command and you can enjoy it. Remember, that you may need to close, and reopen the PowerShell session if you have already used module before updating it.
The important thing is if something works for you on production, keep using it till you test the new version on a test computer. I do changes that may not be big, but big enough that auto-update will break your code. For example, a small rename to a parameter and your code stops working! Be responsible!