Cloud and Edge computing paradigms provide storage and computing services to traditional and Internet of Things devices. One computing platform is not suitable to fulfill the requirements of all IoT devices because of their heterogeneity. In this regard, a federation of various computing paradigms has been emerging, in which a user (first party) with an account on one computing platform (second party) can access the services provided by another computing platform (third party federated with the first computing platform). The user needs to authenticate itself with the third-party computing platform which does not have user credentials. This work proposes a transparent and standard-compliant proxy-based federated authentication for solving the third-party authentication problem in federated cloud and edge computing paradigms. Transparency allows cloud and edge operators to deploy this proxy without having to change the existing authentication protocols. Experimental results illustrate that, as compared with the concatenation of authentication protocols in cloud and edge, proxy-based federated authentication of edge-to-cloud and cloud-to-edge can reduce the authentication delay time by 27.7 percent and 37.9 percent, respectively, and it is also standard compliant.