No, I think you only pay $12.99 if you subscribe via iOS. If you subscribe on Desktop then access it via your iOS device on the same account, you already have a subscription.
I don't know if this is current, but as of 2011, they changed course on In-App Purchases to:
> Content providers may offer In-App subscriptions at whatever price they wish and they are not required to offer an in-app subscription simply because they sell a subscription outside the App Store as well.
At the time, there was a big hullaballoo over providers like Amazon and Comixology who have presumably placed their "best price" on an item, and simply could not afford to take a 30% loss to Apple's IAP. Apple, preferring to have services like Kindle and Comixology vs. not, capitulated.
2011 is (in internet time) a LONG time ago, so it's very possible that this has reversed course again, or even multiple times since then.
Will gladly pay for ad free.
If I pay the subscription fee on my desktop do I have to pay extra because I use iOS? That's not an in app purchase.