I have an iPad mini 2. It is updated iOS 12.5 but most apps require iOS 13. How can I do this update.

Insofar as your iPad is capable, its system software is up to date. The first generation iPad Air, iPad mini2 and mini3 can only be updated to iOS 12.5.5.

Apple ended major update support for these models of iPad in September 2019 - although they since continue to receive occasional “point” security updates - the most recent being released late last year. These devices cannot be updated to iPadOS 13 (or any later major versions of iPadOS) as the internal hardware does not meet the minimum technical requirements (CPU and RAM) for newer versions of iOS/iPadOS.

Some App developers have dropped support for iOS versions preceding iOS 13.x for technical reasons (often because APIs or software libraries upon which they rely have been updated in the new version of iPadOS, making their App incompatible with earlier versions). 

Whilst many App titles remain compatible with your iPad, if you want or need to access to Apps requiring the most recent versions of iOS, you may need to consider replacement of your iPad with a newer or current model that supports iPadOS15.

Some Developers may still have an earlier version of their App available for retrieval from Apple’s App Store servers - but to do so, for new purchases, the App must first be purchased from - and installed to - a device that supports the most recent release of the App. For Users with access to only a single older device, this is often impractical.

I hope this information provides clarity as to your available options - and clarifies the fundamental limitations of your iPad.

An iPad mini 2 isn't capable of 13 or above. Try downloading the applications to a newer unit on the same Apple ID first.

If yours says that a version older than 12.5.5 is up to date, remove the beta profile and restart it.

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