In Cantonese, what is a less "childish" way to call one's own parents? I know the standard ways of calling one's parents—"媽媽/母親" or "爸爸/父親", mother and father respectively, the former more familiar and the latter more formal; or the more familiar/childish way—"媽咪" or "爹哋", mommy and daddy respectively; and of course, the ways I've seen in dramas, the more archaic way, or I suppose the most respectful way—"娘" or "爹", often throwing an "啊" in front for more familiarity. (Please correct me if I'm mistaken in any way)

However, in everyday modern life, how does, for example, a maturing teen, or a thirty-something adult, refer to their parents, either directly and indirectly?

Directly as in calling their parents themselves—"[Mother/Father]早晨!", indirectly as in referring to parents with friends, or perhaps acquaintances, such as "我的 [Mother/Father] 現在不錯".

I've always avoided using any mother/father nouns in indirect speech since, to me, they sound either too rigid (using things such as "爸爸", usually reserved for writing, since this is Cantonese), or too immature (using things such as "媽咪", which to me seems too baby-like, especially for an adult).

So, how would you refer to your parents in a non-rigid and a "mature" way?
In Cantonese, 我爸爸 and 我媽媽 are perfectly fine to use to refer to your parents, no matter what your age is. They are neither rigid nor childish. Slightly more familiar terms are 我阿爸 and 我阿媽. In a very formal context, You can also say 我父母.

To address them directly, most people use 爸爸 and 媽媽, or 阿爸 and 阿媽, depending on which their family prefers.

父親 and 母親 are mainly for writing.