Since you posted this question in the Lightroom for Mobile forum, I will assume you purchase a Lightroom Premium subscription via either the iOS or Android app store. These are mobile-only subscriptions that do not provide Desktop use.
If you have just purchased the Mobile Plan, then yes, you would need to upgrade to the Lightroom Plan to edit on the PC.
The Lightroom Plan normally costs double the Mobile Plan but has 1TB (instead of 100GB) of Cloud Storage and includes the Mobile use also.
I think you can do this via your account but think that a few have had a issue with the files present in the Cloud appearing under the new plan... maybe one of the team will be able to advise the best path.