There isn't really such a thing as "parallel" orbits in the Euclidean sense. On a sphere, parallel lines converge 90 degrees around the sphere.
Suppose you set up a series of satellites to spell out EAT AT JOE'S over Springfield, Illinois. Well, okay, let's make it simpler and just write JOE. For convenience, let's assume the letters are oriented with "up" being north, J in the east and E in the west.
If we consider a single satellite on this orbit, we can find the two extreme points of the orbit: the northernmost point over Springfield; the southernmost point at the antipode, about halfway between South Africa and Australia. There will be two equator crossings, one off the coast of Ghana and one close to Tonga.
So now we run into our problem. If we think about the word JOE made up of a few dozen satellites, the northmost satellite in the J over Springfield (at latitude, say, 40.0001 N) has to become the southmost satellite in the Indian ocean (40.0001 S), and the bottom of the J (39.9999 N) has to become the northmost point (39.9999 S). The letter has to flip upside-down. As the satellite array approaches Africa, the word JOE is going to squish down into a line.
Obviously, if two satellites are precisely aligned east-west and at the same altitude, then logically, they have to pass through each other at the equatorial crossings in order to swap places, which would be a collision. You could fix that by having some of the satellites operate slightly higher or lower, but then you run into the problem that some of the orbits are slightly faster than others, so your letters would smear and come apart over time. So you'd need to carefully position your satellite "pixels" so that any given north-south line contains, at most, only one satellite, all at exactly the same altitude (and it's more complicated than that, because the earth's gravity isn't completely consistent, so it's going to require maintenance -- stationkeeping -- to retain the shape).
So that should answer your question: Any shape you come up with on any set of orbital paths (no matter how close together) has to invert on the opposite side of the planet, which means it will look as intended over one particular area of the planet, be upside-down on the opposite side, and everywhere else it'll be some degree of squished.
Realistically, the point where it "looks right" will move over time based on the difference between the satellites' orbital period and the earth's rotational period, but for this example I'm ignoring that part of the problem. If you put the satellites in a geosynchronous orbit, they'd stay put on the longitude but flatten and then invert every 12 hours (and also move north and south along the longitude line), so in that case it would really be that the image looks correct at a specific time every day and flattened or upside-down the rest of the time.
And that's how satellite constellations work -- they're not a consistent shape that spins around the planet, but a series of interlocking orbits carefully designed so that none of them bump into any of the others. That isn't that hard, you just have to make sure each inclination has a unique altitude, or at least a unique altitude over any given point; we can use slightly non-circular orbits to create completely orthogonal orbits, but that's getting a bit more complicated than I want to go into here.
GPS, for example (from Satellitemap.space):
