You don't use parallax, rather you calculate the direction of the source by using the arrival times of the signals. The nearer detector will receive the signal slightly earlier by an amount that will typically be on the order of a few hundredths of a second. This is essentially the same method that geologists use to determine the epicenter of an earthquake.
This seems horribly prone to manipulation by cosmic radiation, gravity, and a million slower alterations like the rotation of the earth which when combined produce a noticeable effect.
The assumption here is that no force has acted on the energy arriving enough to distort two arrays on earth.
No physicists here but you don't seem to be either. Irc from what had been said with the first event gravitational waves pass through matter and aren't disturbed like em waves, that's why they supposedly open new doors. Secondly, just take it at face value, if they say so it is so, it might seem improbable and "horribly prone to manipulation" but they probably thought of that and have machines precise enough to compensate for whatever effects you or me imagined e.g. "LIGO is designed to detect a change in distance between its mirrors 1/10,000th the width of a proton"[1]. It's like your grandma, never having touched computers, expressing opinions on the next release of some developer tool.
I think the assumption is that the paths taken are almost identical because the distance to the source is vastly longer than the distance between the detectors.
