Okay so first off, no, paper wasn't more durable in earlier periods than paper made the same way today would be. That said, paper was - and is - made in a variety of weights, sizes, and materials, all of which will affect its durability. Paper made purely of wood pulp will not be, for example, as durable as linen or cotton-based papers, but it is generally cheaper (and what we mainly think of today as "paper.")

Manuscripts survive through all sorts of terrible treatment, and parchment and vellum are the most long-lived in wet environments, generally. The earliest paper document from Iceland, for example, is a sales contract from 1437, and the earliest paper manuscript from Iceland, so far, is AM 232 8vo., which dates from approximately 1548. Between then, there are some surviving documents on paper, but not many.

Now, admittedly, paper came late to Iceland - the earliest reference to a paper document is in 1423 - but the lack of even fragmentary sources between the 1423 document and the sales contract in 1437 should give you an indication of the relative fragility of paper.

All that said, you're trying to apply Movie Logic to real life and that...doesn't always work. So far as I'm aware, in the early medieval period in Scandinavia, at least, there were no "bounty boards" or the like to have paper left in the rain. Literacy was not exactly widespread, and there were other, more durable ways of displaying images and text that didn't require expensive materials like paper or parchment. Tree bark, for example, was used in Novgorod, and a famous example is the homework of a young man named Onfim. Those surviving birch barks, though, were found in an exceptionally deep layer and in heavy, waterlogged clay soil, preventing oxidation and deterioration of the material.