I need to update my iPad from 12.5.5 to iPadOS 13.

You don't mention which exact model your iPad is. If it is running 12.5.5, it could be a iPad mini 2 or 3, or the 1st gen iPad Air. If that's the case, as Niel has pointed out, it can't be upgraded past that version. If you need to run a later version of iPadOS, you will need to get a newer iPad model.

Also, note that iPadOS 13 or 14, are no longer available. All newer iPad models now run 15.x.

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’ll need to consider replacement of your iPad with a newer or current model that supports iPadOS15.

Sorry, I appreciate that this may not be what you were hoping to learn.

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

An unit running 12.5.5 isn't capable of 13 or above.

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An iPad running 12.5.5 will always be one of those three. A newer unit would have gone from 12.4.1 straight to 13.0 or above.

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Thanks!