I just clicked on the "Unlock full access" link and after the page with all the pluses and minuses about diamond, platinum and gold, click on the "Upgrade now" and I see a Paypal option along with a credit card option. Maybe it's different in Germany but the option exists for users in the US.
Cant choose Paypal as an option

I can see the option too, by following the steps notmtwain listed.
If for some reason you don't see the option, see this help item:
Additionally, you may send a payment to the order@chess.com address via PayPal and then contact Chess.com Support to let us know your payment has been sent. We will add your premium membership to your Chess.com account.
http://support.chess.com/Knowledgebase/Article/View/139/0/why-does-paypal-ask-for-credit-card-information

Thanks for your reply. The "Try Trial" Button transformed into a "Choose this Plan" Button and now im able to purchase via paypal.
Dont know why it changed now but oh well, im happy it works now.
Edit: Okay so now it keeps telling me something doesnt work right now on the Paypal page. I think i will just send the paypent to order@chess.com and write a support ticket so you guys can add my membership.
Hello everyone,
i wanted to purchase a diamond premium account via the membership page but there is only the option to begin a trial. The page keeps asking me for my credit card information but i dont have a credit card.
Yesterday, there was some kind of countdown - "just 10 hours left to get your trial" - so i thought maybe if this period of time is over, i will be able to purchase a diamond membership via paypal but now, far more than 10 hours later, there is still just the option to get a trial version where i HAVE TO pay with a credit card again.
I already wrote a support ticket but did not get an answer yet. Maybe someone could help me out here. Also, there were some topics about this issue in the past but no solution it seems.