开一个生日会 explanation as to why 开 is used here?
开会 (“to host/attend a meeting”) is an example of a separable verb—we can add content between the two characters. The 开 relates to the “hosting” whereas the 会 relates to the “meeting”, and the in-between content modifies the 会.
For example, we might say:

他在开一个会。  (“He is hosting a meeting.”)
我们周五开一个三个小时的会。  (“On Fridays, we host a 3-hour meeting.”)

In the given example, instead of a 会(“meeting”), it’s a 生日会 (“birthday party”), but the same idea applies.

开一个生日会 (“to host a birthday party”)
爸爸开儿子的生日会。 (“The father hosted his son’s birthday party.)

Alternatively, we can think of 开 itself as an independent verb.