Poster paint is also known as Tempera paint (in the US, anyway), though not the same thing as Egg Tempera (which you must not use as it does often contain toxic pigments), and is considered a good paint for younger children due to its water-based nature; it tends to be easier to clean up, and is available in "Non-Toxic" formulations. It's often recommended for use for "baby footprints" and such, though I can find no scientific source for this - just lots and lots of parenting articles, which suggest it's popularly considered safe, but of course that is not always accurate.
There are formulations of Tempera/Poster paint that list ages as young as 18 months, such as this one (no endorsement of the product implied or meant). Often age listings on products have more to do with what the product manufacturer is willing to test for, but if you're able to find a paint that lists 18 months you could certainly be more confident in that than one that lists 3 years; it's not possible to say if a 3 year approved paint is okay for a 1 year old, given (as mentioned by @anongoodnurse in comments) there's a lot more likelihood of paint going in the mouth.
If you want more information on what's in poster/tempera paints, you can read this article; it explains that there are relatively simple ingredients (water, calcium carbonate (chalk!), binding agent (usually a starch), pigments, and preservatives). The pigment is the big question mark safety wise - most likely they would be food-grade, but it's just a guess, as manufacturers protect that information as trade secrets. Look for the certification of non-toxic (not just the words, but certified by a government/NGO body that is well respected; in the US there is the "AP" seal for example).
The nice thing about tempera paints is that their simple ingredients lead to a better option: make it yourself. See for example this recipe, though there are hundreds out there; all it takes is corn starch (a.k.a. corn flour in the UK), water, and food coloring. (The calcium carbonate is not necessary, as it makes the paint more matte - but toddlers and infants don't typically care about that!). It's also probably a lot cheaper than poster paint or tempera paint. Just make sure that you don't keep it around for long - just like the soggy bread that it more or less is, it will spoil relatively quickly without the preservatives found in commercial stuff!