What you are asking for is technically possible, and other answers have already gone into the details about how you could accomplish this.
What I'd like to go over in this answer is how you probably should architect this given the requirements that you described. Keep in mind that what I am describing is one way to do this, there are certainly other correct methods as well.
Create a database on the server where your app will live. A simple MySQL DB will work, but you could use anything. Create a table called magazine, with a column url. Your code would pull the url from this DB. Whenever the magazine URL changes, just update the DB and the code itself won't need to be changed.
Your front-end code needs some sort of way to access the DB. One possible solution is a REST API. This code would query the DB for the latest values (in your case magazine URLs), and make them accessible to your web page. This could be done in a myriad of different languages/frameworks, here's a good tutorial on doing something like this in Node.js and express (which is what I'd personally use).
Finally, your front-end code needs to call your REST API to get the updated URLs. This needs to be done with some kind of JavaScript based language. jQuery would make this really easy, something like this:
$(document).ready(function() {
$.Get("http://uri_to_your_rest_api", function(data) {
$("#myImage").attr("scr", data.url);
}
});
Assuming you had HTML like this:
<img id="myImage" src="">
And there you go - You have a webpage that pulls the image sources dynamically from your database.
Now if you're just dipping your toes into web development, this may seem a bit overwhelming. But I promise you, in the long run it'll be easier then trying to parse code from an HTML page :)
webscraping. On the backend, something like that is quite easy to implement, but on the frontend it's rather "hacky" and inefficient.