I was pretty curious to try it myself, so I created a test application and I was easily able to get a postback within the IFrame.
Now a little clarification if a postback is happening inside an IFrame, you will even see the effects within the frame bounds (unless programmed other wise using some client script). Now to your issue there is no possibility of you invoking the parent __dopostback
event, under normal circumstances (but you may like to see this link for a probable thing that might be happening at your parent page).
At times as a security measures the javascript is disabled on individual IFrames (the feature is readily available with HTML5). There is a good possibility that you may be dealing with a similar situation. In that case you do not have any control over the functioning, unless you fall back to simple HTML forms and post it to a different ASP.net page.
Based on our discussion below I would suggest you to use simple HTML hyper link in your page, unless you can change the code of the parent page. If postback is desirable then you can make use of cross page posting (as I already mentioned) and from that page redirect back to your current child page.
One of these codes should work for you:
<a href="#" onclick="YourFormName.submit()">Link Text</a>
OR
<a href="#" onclick="document.getElementById("myForm").submit();">Link Text</a>
OR
<a href="#" onclick="document.getElementById("Iframe Name").contentWindow.document.getElementById("YourFormName".submit)">Link Text</a>
OR
<a href="#" onclick="document.getElementById("Iframe Name").contentDocument.document.getElementById("YourFormName".submit)">Link Text</a>
If none of the above codes works for you then you should consider putting a submit button instead of hyperlink.
Hope this helps.