why is chicken not parve? Rabbi Jonathan Ginsburg

The source of the prohibition against consuming a mixture of meat and dairy is the verse, "You shall not cook a kid in his mother's milk" (Exodus 34:26). This verse infers that only the flesh of mammals, whose mothers lactate, are included in this prohibition. Fish and poultry do not nurse their young, and are thus excluded from this prohibition.
Nevertheless, rabbinic decree forbids the consumption of poultry or fowl together with dairy products. This injunction was enacted because their meat can be confused with beef, and people may not know to differentiate.
The prohibition mentioned in the Torah is to mix meat (beef) with milk. This injunction was extended by the Sages to include fowl as its meat can be confused with beef. Eggs do not fall into this category as they cannot be mistaken for meat. Furthermore, eggs are considered a separate entity once they have been hatched and are considered to be pareve, or neutral, so that they can be eaten with either milk or meat.