I can say that obviously the planets of other stars do not have orbits that are either 'in" or "parallel to" the galactic plane.
And I explain why below.
If you look at maps at the sky you will see that realtively nearby stars are see in in all directions from Earth. Or you could go out on a clear night at a place with an unobstructed view of the sky and see stars in every direction in an entire hemisphere of the sky.
Earth is in the disc of the galaxy, which is usually described as being about 100,000 light years in diameter and about 1,000 light years thick, though it's edges are very vague and fuzzy.
Stellar density is the average number of stars within a unit volume. It is similar to the stellar mass density, which is the total solar masses (MSun) found within a unit volume. Typically, the volume used by astronomers to describe the stellar density is a cubic parsec (pc3).
In the solar neighborhood, this value can be determined from surveys of nearby stars, combined with estimates of the number of faint stars that may have been missed. The true stellar density near the Sun is estimated as 0.004 stars per cubic light year, or 0.14 stars pc−3. When combined with estimates of the stellar masses, this yields a mass density estimate of 4×10−24 g/cm3 or 0.059 solar masses per cubic parsec. The density estimate varies across space, with the density decreasing rapidly in the direction out of the galactic plane.1
The locations within the Milky Way that have the highest stellar density are the central core and the interior of globular clusters. A typical mass density for a globular cluster is 70 MSun pc−3, which is 500 times the mass density near the Sun.2 In the solar neighborhood, the stellar density of a star cluster must be greater than 0.08 MSun pc−3 in order to avoid tidal disruption.
Imagine a cylinder of space including the Solar System, and being 1,000 light years "high", and perpendicular to the galactic plane, going from the "top" to the "bottom" surfaces of the galactic disc.
Suppose that such a cylinder has a radius of 10 light years. It will have a volume of 314,159.265359 cubic light years and should include about 1,256.6 stars, including the Sun.
Suppose that such a cylnder has a radius of 5 light years. It will have a volume of 78,539.816340 cubic light years and should include about 314.16 stars, including the Sun.
Suppose that such a cylnder has a radius of 1 light year. It will have a volume of about 3,141.592654 cubic light years, and thus probably about 12.5 stars, including the Sun.
Thus it is apparent that the disc of the Milky Way Galacy is many star systems "thick", and that the stars do not all orbit in exactly the mathematical central plane of the galaxy. Instead stars have obits which are slightly tilted relative to that plane, and so they spend half of each orbit around the galactic center "above" the plane and half of each orbit "below" the plane.
So at any moment of time only a tiny fraction of stars could be at the galactic plane. Thus it is not possible for more than a tiny fraction of stars to have their planets orbiting in the galactic plane at any one moment.
Could the planes of all the planetary orbits of all the stars in the galaxy be paralle to the central plane of the galaxy?
Most sky maps use the equatorial coordinate system, with the poles above the North and South poles of the Earth, and the celestial equator above the equator of Earth.
But the Earth's equator is tilted from the plane of Earth's orbit around the Sun, the ecliptic plane, by about 23 degress. Since the orbits of the other planets are tilted by no more than a few degrees from the ecliptic plane, an ecliptic coordinate system is used to track the apparent movements of the other planets around the sky, and that system has a central plane titled by about 23 degrees from the central plane in the equatorial cordindinate system, and the poles are 23 degrees from the poles in the equatorial system.
The north ecliptic plane is at about 18 hours and + 66 degrees 33 minutes in the equatorial coordinate sytem, and the south ecliptic pole is at about 6 hours and about - 66 degrees 33 minutes in the equatorial system.
The ecliptic plane passes though the constellations of Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Ophiuchus, Sagittarius, Capricornus, and Aquarius.
Astronomers also use a galactic coordinate system to plot the directions to distant stars and objects.
The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the Milky Way Galaxy, and the fundamental plane parallel to an approximation of the galactic plane but offset to its north. It uses the right-handed convention, meaning that coordinates are positive toward the north and toward the east in the fundamental plane.1
The north galactic pole is at about 12 hours 51.4 minutes and + 27.13 degres in equatorial coordinates, in the directin of the constellation Coma Berenices. The south galactic pole is at about 0 hours 51.4 minutes and - 27.13 degres in equatorial coordinates, in the constellation Sculptor.
The glactic equator in the galactic coordinate system runs through the constellations Sagittarius, Serpens, Scuta, Aqulla, Sagitta, Vulpeca, Cygnus, Cepheus, Cassiopeia, Camelopardalis, Perseus, Auriga, Taurus, Gemini, Orion, Monoeros, Canis Major, Puppis, Vela, Carina, Crux, Centaurus, Circinus, Norma, Ara, Scorpio, and Ophiuchus.
So by comparing the polar coordinates of the ecliptic and galax ctic coordiante systems, and the constellations that their equators run through, you can see that the plane that the planets in our Solar System orbit in is title by a large amoung from being parallel to the galactic plane.
Could most star systems have planetary orbits parallal to the galactic plane and our Solar System be a rare exception?
In our solar system, the planes of the orbits of the planets are tilted from the plane of the Sun's rotational equator by no more than a few degrees. And modern theories of planetary formation indicate that it would be usual and typical for planets to form with orbital planes near the plane of their star's rotational equator.
Planets whose orbital planes are greatly different from those of their stars should be rather rare. And as far as I know the rotational axis and planes of stars are randomly distributed.
Here is a sky map showing the positions of stars with known exoplanets:
The map is arranged in equatorial coordinates, but the Milky Way is marked on the map, and thus it can be seen that the stars with known eoxplanets are scattered all over the sky, including in directions which ae far from the galactic plane.
And here is a link to a list of constellations with the number of known exoplanets in them:
This shows that stars with known exoplanets are scattered all over the sky as seen from Earth.
And the usual methods for detecting exoplanets make it much easier to detect exoplanets orbiting their stars in orbital planes near to the line of slight between them and the Earth. Most of the 4,000 plus exoplanets detected so far have orbitalplaneswhich are not titled by very many degrees from the lines of sight between them and the Earth.
And that includes exoplanets orbiting stars in directins new to the galactic poles as seen from Earth. The orbital planes of those planets are close to being perpendicular - at right anges - to the galactic plane.
So the evidence indicates that the majority of planetary systems in our galaxy do not hve planetary orbits parallel to the galactic plane. Instead the planes of planetary orbits are orientated in various random directions.