Because the brick around the door is sheltered by the overhang you could probably paint it without having to worry about damaging its ability to breathe. However, there is an alternative way to change the color of brick and mortar. I've been fascinated by DYEBRICK since I first discovered the company. The non-toxic, water-based stain soaks in so the texture of the brick is retained. In addition to traditional brick colors they offer LimeLike that can significantly whiten even dark brick. Final color is determined by dilution rate with the fixative. HERE is a photo of one job. The only problem is that people from the U.S. and Canada have to order the product from England.

I'm not sure what Durock is but we want white around the red door. I'll google durock. Here is sort of the look I had in mind (though this would not be possible on our home).

You can take brick molding off from the door, build it out and use Fypon surround or something similar. Here is a link, you can take a look and see if you like anything.

I would paint it a soft shade of white & not worry about maintenance since it’s recessed from the elements. For such a small amount of brick I wouldn’t think maintenance would be an issue.

I agree with K Laurence. It's such a small amount of brick and recessed. Paint it.

some ideas

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Personally, I think the brick looks nice and normally do not like painted brick. However, for such a small amount, paint it or cover it as suggested above or rip it out and apply some other siding. I'm assuming it's just brick veneer. If you did that, you could probably fit in a door with one side lite.

Just to be clear, are you also painting the sidewalls of the recess the grey or white like your inspiration photo? If you are painting the sidewalls grey, I would paint the brick wall grey. Don't add stone, stone and brick don't pair well together, especially if the stone is not repeated anywhere else. Same with wood, don't do that. A stucco wall instead of the brick would work. Have you considered a white door, or a lighter coloured door to brighten this entrance? The LRV number is 8.12 for BM Pomegranate, that's going to look very dark in a recess. Have you thought about a brighter color for the door?