Last updated 2020/03/04
Some links from twitter on the topic of parsing PSD files (haven't looked into them in details). PSD include a flattened rasterized image so this is easy to load if that's the only thing you need. Most software / librairies have support for extracting this flattened data (e.g. stb_image.h does).
However if you want access to individual layers, render non-rasterized layers, emulate every photoshop features, extract or apply effects with more granularity, more code is needed. May range from easy to lots-of-work depending on what exactly you need.
As far as I know there isn't a trivial stb-like ready-to-use C++ library to do that sort of things. Posting all links here. Some are probably bloated or hard to use into your project, some lacking features.
TODO: Actually look into the pros/cons of all those.
Adobe Photoshop File Formats Specification (October 2013)
@donzanoid "this guy goes round the houses to use that source to do what you want, export layers to png"
@donzanoid "slightly simpler (one c/h) but not as feature complete"
Molecular Matters' psd library SDK
(possibly the best of this list, comes with sources, but at a cost)
Molecular Matter's psd extract tool
March 2020: this is now open-source: https://github.com/MolecularMatters/psd_sdk
Paint.NET PSD Plugin