Historical linguist here. This type of question is at the very heart of what we do. However you're really asking the million dollar question.

We simply don't know the answer. There's certainly no evidence to support that they all evolved from a single parent language, but that's just because we don't have records that go back that far. We can reconstruct parent forms, like Proto-Indo-European or Old Chinese, but these reconstructions are heavily reliant on having adequate data. Every language that's ever died is then a missing piece in the puzzle.

There are people who've proposed massively include families like Nostratic (not widely accepted) or Sinot-Tibeto-Austronesian (not widely accepted) but these generally lack enough support. Even much smaller pairings like Koreo-Japonic don't have the support to be widely accepted by the people who would really know.

We currently are not able to know if languages developed independently or not.

tl;dr: No one knows because it's not currently knowable.

Not a linguist, but I think I can point you in the right direction.

Did languages develop independent of each other?

There are a few languages that, as far as we know, are not related to any other known language. These are called language isolates. Korean is probably the best known example, followed by Basque.

Or do they all come from one language?

Unlike isolates, most languages evolved from a common ancestor. More-or-less all of the languages of Europe, central Asia, and south Asia came from a common ancestor called proto-indo-European, which would have been spoken about 5000-7000 years ago. Linguists can reconstruct what this language was like by looking at words and sounds that are common across all of these languages.

Going back even farther, linguists can attempt to extrapolate what the very first spoken human language might have sounded like: https://en.wikipedia.org/wiki/Proto-Human_language