How can I pass a List of Ids as a parameter using an ActionFunction?

Actually you cant send a list using actionfunction. To pass a list to controller you can consider using Remoting(Stateless).

If you want to send a list of Ids you can always send a csv formatted string using js Something like

<script>
    UpdateSelected('a1we00000004EXAAA2,a1we00000004EXBAA2');
</script>
<apex:actionfunction name="UpdateSelected" action="{!TestAction}" rerender="r"> 
    <apex:param name="ids" value="" assignTo="{!SelectedIds}" /> 
</apex:actionfunction> 

and in the controller method you can split the string into list by

Example :

public String SelectedIds{get;set;}
public void TestAction(){
 List<String> allIds = SelectedIds.split(',');
}