The Torah said, “Whether it is a bull, a sheep or a goat, do not slaughter [a female animal] and its child on the same day.” This prohibition is true both in the Land of Israel and outside, in the time of the Temple and now, for ordinary animals and for sacrifices.
For example, if one slaughtered an animal and its offspring, both regular animals, outside the Temple, and both are now fit for consumption, then for the second slaughter he is liable like for any other negative commandment of the Torah. However, if both the parent and the offspring were sacrifices and he slaughtered both outside the Temple Courtyard, then for the first slaughter he is liable because he did it outside the Temple, and may be cut off from the people, but the second animal is now unfit for slaughter on the same day, and thus the slaughterer is not liable for slaughtering it outside. He is liable for "it and its offspring" slaughter though.
Two different people can slaughter the two animals, and the second one – that one who does the action now forbidden – only he is liable.
Art: Eugène Verboeckhoven - Shepherd Girl With Cattle And Sheep At Rest