Not the parent, but I would definitely recommend it. A lot of voice quality has to how you pronounce vowels and where you place the resonance in your mouth. Also, avoiding singing through your nose (hold your nose closed and if there's pressure, then don't sing that way). There are vocal exercises for all of those, as well as exercises for breath support, singing in tune, phrasing, etc.

If you'd like to improve, definitely! There are all sorts of things involved in singing that some do naturally and others (like myself) do not. Having a knowledgeable teacher who's able to observe you, identify areas of improvement, and help you progress is really beneficial. There's a whole lot of "not knowing what you don't know" at play.

Plus singing is a pretty personal way of making music, and there can be a lot of social fear of doing it badly. Formally studying it can help with getting over the psychological hump from self-identifying as "someone who is attempting to sing" to "singer".

I think you should pursue anything you're interested in.

I don't have the best singing voice, but after a few years of learning how to sing, it's like I finally "get it". It used to be this very mystical thing, but now I realise how to use my body to sing properly.

I would temper your expectations. Singing lessons will improve your singing faster than practicing alone but it still may take years and years to develop.

I had always enjoyed singing and I started singing and piano lessons at the same time. With consistent practice you can improve at the piano, learn more complex pieces and see your progress. With singing the process is a lot more mysterious. You probably don't understand the things that are going on inside your body very well and the teacher can't just say "Try putting your first finger on this key and your middle finger on this key".

After three years I could play reasonably complicated pop songs on the piano and read chords. I'm still a few steps the average "someone who likes to sing level".