Learn Finnish: Uusi käsikirjoitus on parhaillaan tekeillä. - A new manuscript is currently in the making.

tekeillä means "in the making" or "in the works". It is built from the verb tehdä (to do/make) using a noun-like stem teke- and adding the plural adessive ending -illä. It acts as an adverbial showing that the subject is currently undergoing a process.

The lesson mentions these words often take a possessive suffix. Why is there no possessive suffix on tekeillä here?

When the subject of the sentence is the inanimate thing being acted upon (like the manuscript), words like tekeillä or suunnitteilla (in the planning) do not take a possessive suffix. They simply describe the state of the object. If you added a possessive suffix here, it would imply the manuscript itself is actively doing the making, which doesn't make sense.

parhaillaan looks like it has the same grammatical structure as the lesson topic. Is it related?

Yes! parhaillaan is formed from the plural adessive of the adjective paras (best), which is parhailla-, plus the third-person possessive suffix -an. While its literal roots mean "at its best", it is used as a fixed adverb meaning "currently" or "at this very moment". It pairs perfectly with ongoing process adverbials.

Could I use a standard verb structure like the third infinitive instead of tekeillä?