Hi i know its kind of late but, today i had to do the same thing, so i followed a great answer on sharepoint.stackexchange.com. But basically is like adding any other asp.net control on an APS.NET page.
First you need to define the prefix for that assembly
<%@ Register tagprefix="prefix" namespace="NameSpaceOfTheWebPart" assembly="FullAssemblyName" %>
ex:
<%@ Register tagprefix="csm" namespace="MySolution.VisualWebPart" assembly="MySolution Version=1.0.0.0, Culture=neutral, PublicKeyToken=335eef5f6f60e56b" %>
and then add the web part to the page (on the main place holder) using the required attributes by asp.net (ID and runat).
<prefix:WebPartClassName ID="ID" runat="server" />
The only thing is that you won't be able to edit the page and modify the Web Part through the browser, if you need to modify any other property of the Web Part you will have to do it from the html or code behind.
here is the url of the link i followed: https://sharepoint.stackexchange.com/questions/26508/is-it-possible-to-add-web-parts-to-an-application-page
i hope it helps.