To do this you will want to send the initial two calls to a conference, and then have your new web page also connect to that conference.
https://www.twilio.com/docs/api/twiml/conference
If this happens a lot, you can just have all of your calls running through conferences, and keep a list of conferences that can be joined by your new web page.
If this is infrequent, you can use the rest api to modify the calls in progress:
https://www.twilio.com/docs/api/rest/change-call-state#redirect-a-running-call-to-a-new-url
Grab the SIDs, redirect them to a URL that dials a conference, then join that conference with your new web page.