On a wide angle lens hood petals should be open wide enough not to obscure field of view; on a telephoto lens narrow petals would block sun better if shooting close to the sun. I imagine it would be useful on a zoom lens to have a hood with petals that could move wider when zooming out and narrower when zooming in. Does anybody manufacture something like that?
There are a few zoom lenses that effectively do what you envision, but it is not due to movements of the hood. Rather, it is due to a retrofocus design that extends the lens barrel fully at the widest focal length and retracts the lens barrel fully at the longest focal length. The hood, rather than being attached to the front barrel that extends and retracts, is attached to the main barrel that remains stationary.
The original Canon EF 24-70mm f/2.8 L is one such lens.
With the lens set at 24mm, the front element is extended almost to the end of the hood attached to the main barrel. This gives wide angle coverage with the hood.
With the lens set to 70mm, the front element is retracted all of the way into the main barrel. If the hood were in place, this would give much narrower angle coverage with the same hood. Notice that the attachment tabs for the hood are on the main barrel, rather than the inner barrel that extends.
The Zoom Nikkor AF 35-70mm f/2.8D had a similar optical design. The lens was at 35mm fully extended and 70mm fully retracted. But the push-pull zoom action that Nikon gave the lens prevented the hood from being attached to the main barrel. As a result, the hood was a thin ring attached to the moving front of the lens that was hardly worth the trouble.
No. There are several (probable) reasons why they don't exist
Unneeded complexity. Pros using zooms (such as the 14–24 mm, 24–70 mm, and 70–200 mm ƒ/2.8 trinity) are moving and shooting fast. The last thing they want is to ruin a shot when they zoom out to the lens's widest angle, but forget to tweak the hood's angle, and get vignetted/cropped images.
Purpose. Many photographers use a lens hood as much as for front-element protection as they do for light blocking. A fixed plastic guard attached to the front of a lens provides a simple "no-go" zone to prevent objects, people, branches, etc., from getting too close to the front element (much like a US football player's facemask). A hood with 4+ hinged petals might break, or at least just bend easily out of the way from protecting the front element.
Cost. In order to be useful, 3rd-party "variable"-petal hoods would have to be custom made for each lens. Manufacturers all use different hood attachment mechanisms, and there are different hood sizes for different lens barrel diameters. So the 3rd-party market wouldn't benefit very much from one-size-fits-all product design.
Conversely, they could make them mount to the lens's filter threads. But that starts to create its own problems, such as needing a rotation mechanism to make sure you can adjust the petal orientation to match the camera body orientation. It also starts adding stack depth to the lens filters. Add a ND or polarizer to the filter stack that starts with the "variable" petal, and suddenly a filter that previously caused no problems now shows some vignetting issues.
So what about the lensmakers themselves? Well, pros don't seem to be clamoring for such a feature, so high-end lenses probably wouldn't get them. And consumer-grade and entry zooms probably won't get them because the more complicated hood would add to the product cost (in manufacturing, and more importantly in product support to deal with broken flexible or hinged mechanisms). The support cost is probably higher than the manufacturing cost.
All the other drawbacks. Not as fast to remove, turn around, and reattach to the lens when putting it in the bag. 1st-party variable hoods would be the least bulky, but are certainly bulkier than the current wide-angle only hoods on zooms. 3rd party variable hoods, probably meant to work with multiple different lenses, would certainly be bulkier.
Added weight to the front of the lens. In order to prevent the top petals from sagging into the field of view, the hinge needs to be stiff enough to hold up the weight of a cantilevered petal, but compliant enough to allow the photographer to move it easily when desired. That requires more plastic ribbing or reinforcement to prevent stress cracks at the hinge jaws. (And yes, I'm assuming they'll be made of plastic, for weight and cost reasons).
Now, to get similar functionality, without being strictly a variable-petal hood, there are a couple options:
3rd party rubber hoods. They can be withdrawn (usually in 2-4 segments, similar to silicone collapsible kitchen bowls, measuring cups, etc.) to provide a wider angle of view.
But they're round, not petal-shaped, so if you're concerned about keeping the hood as close to the field of view as possible, round hoods are suboptimal.
Barn doors on matte boxes. They're used a lot in videography (they're most often mounted as part of a video rail system, rather than directly to the front of the lens).
I've never heard of a photographer using mattes and barn doors, but there's nothing preventing you from doing so (besides cost and convenience, that is).