Is there a way to give someone else permission to locate my phone for safety reasons? My father wants to be able to locate my Android phone (a Galaxy S4, fully updated with Lollipop) in a similar way as how he can locate the iPhones he and my mother use, for safety reasons in case something happens and he needs to know where I am.

I do agree with this, but I would rather not share any of my passwords with him. This means that the apps that I currently use, like Lookout, my Google account or my Samsung account are not really options, unless these apps have some kind of family sharing function. I would also very much prefer a free option. I found some options, but they're either paying or would require account sharing.

I would prefer a method where I can load an app on my android phone that I connect to using my Apple ID, after which my father can use the built-in Apple ID family options to track my phone, but if that's not an option, I'm very much open to any other suggestions.
I stumbled upon this nice link where some possibilities are listed. 

From your question I derive that your father is an iOS user or at least familiar with apple services, so the cross platform apps might be of special interest for you. For some approaches, you can even define some kind of a geo-fence where people get informed when you arrive within these coordinates. For example, your father can be notified as soon as you arrive home after school. 

But please keep in mind that if you share your location with other users, your GPS location is uploaded to a server where the other users get their data from. Depending on the service's provider, the connection and storage can be secure or not and the data might be used for eg advertisements or not. So please have a close look before deciding. GPS coordinates are by design privacy-critical information, so while it might be ok for you sharing your location with your family, it might not be ok for you to share it with companies that might monetarize the data.