Given the angular diameter $a$ in radians and the distance $d$ in Mpc, you can get the actual diameter $D$ from:
$$D = d\tan{a}$$

Using the small angle approximation, you get:
$$D = da$$

$a$ is in radians, so to get the distance in Mpc from the angular diameter in arc seconds you'd need to convert the angle in arc seconds to the angle in radians: $a = \frac{2\pi A}{360\times3600}$, where the factor $3600$ is used to convert arc seconds to degrees, and $\frac{2\pi}{360}$ to convert from degrees to radians:

To get the diameter in kpc:
$$D = 1000\times d \frac{2\pi A}{360\times3600}$$

$$D = d \frac{\pi A}{648}$$
$$D \approx \frac{dA}{206} $$

where $d$ is in $\textrm{Mpc}$, $D$ is in $\textrm{kpc}$, and $A$ is in $\textrm{arcsec}$.

Here $D$ is the diameter for round objects (even for disk galaxies seen at an angle). If the object is not round then this would normally be the maximum diameter.

[EDIT (see answer and comments from HDE 226868)] For irregular galaxies you may also need to have more information (viewing angle) to find the real maximum diameter of the galaxy. But that information is (I think) only available for galaxies in the local group.

I'm not sure I understand the second formula (could you explain the variables?), but I do know that you can use parallax in certain situations. Parallax works by using the idea that objects appear to be in different places when you (the observer) are at different places - i.e. at different points in the Earth's orbit. This graphic gives a good description of the process.

Credit: Wikipedia.

What I think your friend is trying to get at is that if a galaxy is tilted at a certain angle, it will appear foreshortened, and it will seem smaller than it should at that distance. This could lead you to make an incorrect measurement of its angular diameter and therefore an incorrect measurement of its actual diameter. You could correct this using trigonometry if you knew just at what angle the galaxy was tilted, but it would be hard to measure that angle. You could avoid this whole dilemma if the galaxy was perfectly circular, because you would always see the longer side even if it was foreshortened, but that could be an invalid assumption.

By the way, the reason this isn't a problem with ideal, perfectly spherical objects is that they are perfectly spherical and cannot appear foreshortened.