Any finite poset has discrete (therefore Hausdorff) interval topology. And the image of any finite poset is finite, so the question has a positive answer for finite posets. I suspect the answer to the question is positive for all posets, but I don't know.

That's right - > just edited

Thanks - this scenario escaped me... pls post as answer, I will tick it!

That's right, Ramiro, and thanks for your remark. My question should have been: is there any reason why this sharper version (ultraparacompactness as opposed to paracompactness) doesn't work? Maybe ultraparacompactness is also open...