This could also be done in Illustrator (inspired by the answers here):
If the original artwork is vector...
1. Convert your artwork to a Symbol
Put your artwork on your main artboard. Select all of the artwork, then hit the New Symbol button on the
Symbols pane (alternatively you can drag the selected artwork to the
Symbols pane). That will convert your artwork to a symbol.
2. Create a set of artboards, one for each output size you want.
I've got 3 artboards of varying sizes. Each just has the symbol that I've made from the SE logo. Then, from the
Symbols pane, drag out your artwork and place it on the other artboards (and resize it as you like)
3. Edit the logo as needed
To edit the logo, just double click the symbols on the
When done editing, hit escape and all the instances of the symbol will get updated:
4. Export the artwork
The exporting part is where Illustrator's power play comes into effect.
Export... as PNG, make sure you
Use Artboards (all):
The artboards and your resolutions will all be automatically saved based on the artboard name.
If the original artwork is from Photoshop...
Very similar set up, just ignore the symbols part. Instead, use
Place... and place the .PSD into the document as many times as you have artboards. You can still update the artwork in Photoshop. Every time you save it in PS, Illustrator will ask if you'd like to update it:
However, at this point I'd probably go with Yisela's solution instead. No need to split the workflow up between two different programs.