How did astronomers determine the path of 'Oumuamua so quickly? The interstellar asteroid 'Oumuamua was discovered on October 19, 2017. Once discovered, less than a month's worth of precovery observations were found on various sky surveys. By mid-November, scientists had worked out its orbital eccentricity, and soon after its probable origin and destination.

With such a quick visit and so little data, how did scientists figure out so quickly where it came from and where it was going? Did using space probes for observations impart enough parallax to easily determine that, or did Earth-based observation have enough parallax themselves?
Three accurate observations are sufficient to fix a Keplerian orbit (ie an elliptical or hyperbolic orbit with the sun at the focus)

In practice, observations are not perfectly accurate due to limitations of the equipment and observations over a short time are particularly prone to observational error being magnified. Moreover the orbit will be perturbed by the gravity of the planets, so won't be perfectly Keplerian. For this reason pre-discovery images will help fix the exact orbit more accurately.
As the time difference between first and last observation is critical in the quality of the orbital determination, we talk about the length (in days) of the observation arc as a measure of how well defined the orbit is.

Multiple observations can reduce error by an averaging effect (the Gaussian curve is so named from Gauss's use of it in orbital determination) 

However once you have three or more observations of a body you can determine its orbit almost immediately, using a computer to do the calculations for you. It would have been immediately obvious that the object was on a very unusual trajectory.

There is no need to use space probes for the observations. The relative motion of the Earth and the Object provide sufficient parallax to determine distance, position and velocity.