The phenomenon of parallax itself is simply the result of different views of one's surroundings observed from different locations. For binocular vision, we get a different view of our environment from each eye simultaneously, and the visual cortex of our brains learns very early on how to process the two distinct images into an appearance of a three-dimensional image. (Actual physical interaction with the objects in the environment is also important in "educating" the visual cortex about distances to objects. We generally accomplish this during infancy.)
In the case of astronomical parallax, celestial objects are generally so distant that the two (or more) different views must be obtained at different times from different positions of the Earth on its orbit. (The exception is the Moon, which is close enough to show parallax in simultaneous observations from well-separated points on the Earth. As an example, this is why occultations of stars by the Moon are not seen by all observers on the otherwise proper side of the Earth.)
To note the presence of astronomical parallax, it is necessary to have very precise measurements of the direction in which the object is seen, relative to a much more distant "background". The fact that this is not observed with the naked-eye was used until the 18th Century as an argument against the motion of the Earth about the Sun, since (obviously) the planets and stars would show parallax if the Earth were not simply stationary at the center of the Universe.
One difficulty in measuring parallax for stars is that, as we now know, the change in observed direction, even at diametrically opposite locations on the Earth's orbit, is at most a bit over one second of arc (1/3600 of a degree) for the closest stars and generally hundredths of a second or less for most of the visible stars. But even to discern that requires being able to compare the observed directions of the stars at different times. In the age before photography, this required having precise charting of the stars on the sky, which was the result of a tremendous effort in the centuries after the first use of telescopes in astronomy. It wasn't until 1838 that the first sufficiently accurate observations of stellar parallax were accomplished.
By comparison, the aberration of starlight due to the Earth's motion is an effect about twenty times larger than the largest stellar parallaxes, so it was became possible to detect that by 1725.