On a already created webzone I have two web parts, that are arranged one under the other. I want to arrange them side by side but I don't have admin permission or SharePoint Designer. how can I do that?
You may be able to create a personal view of the web part page. This gives you the ability to re-arrange the web parts the way you want and not affect other users.
On the top right hand side of your website, click the dropdown that has the name of the site in it. At the bottom of the selection you should see "Personalize this Page".
Here's an article with a bit more detail: https://answers.uchicago.edu/page.php?id=25428