DateTime.Parse(txtCalendar.Text);

This means that you need to write a full translation class to pass date format strings to the calendar control in order to be culture-aware.
This sounds problematic, and is apparently a lot more work than it sounds. Did anyone make something like this yet?

the
Sys.WebForms.PageRequestManager.getInstance().add_endRequest(EndRequestHandler);

worked perfect!!

function EndRequestHandler(sender, args) {
    if (args.get_error() == undefined){
        renderCalendarControls("updpcalendar");
    }
 }
function renderCalendarControls(rel)
{
    var textBoxes = Ext.DomQuery.select("input[rel=" + rel + "]");
    Ext.each(textBoxes, function(item, id, all){
        new Ext.form.DateField({
        applyTo: item
        });
    });
}
Ext.onReady(function(){
    renderCalendarControls("calendar");renderCalendarControls("updpcalendar");
    Sys.WebForms.PageRequestManager.getInstance().add_endRequest(EndRequestHandler);
});

For the text boxes within update panel, I used rel=”updpcalendar” and for the other, I used rel=”calendar”

I have not tested the following code, but hopefully it will help

function EndRequestHandler(sender, args) {
   if (args.get_error() == undefined)
       renderCalendarControls();
   else
       alert("There was an error" + args.get_error().message);
}

function renderCalendarControls()
{
	var textBoxes = Ext.DomQuery.select("input[rel=calendar]");
	Ext.each(textBoxes, function(item, id, all){
		var cl = new Ext.form.DateField({
			allowBlank : false,
			applyTo: item
		});
	});
}

Ext.onReady(function(){
	renderCalendarControls();
	Sys.WebForms.PageRequestManager.getInstance().add_endRequest(EndRequestHandler);
});