働く and 勤める: work, function, and an employment relation
Compare a person working and a spring functioning with employment at a library and a father's former factory, keeping subjects and current or past timing distinct.
働く: human work and a spring's function
働く (はたらく) means to work or to function. 「兄は駅の近くの会社で働いている。」 means “My older brother works at a company near the station.” The subject is a person working at a company. 働いている describes the older brother's current work there, without supplying a particular role or employment terms.
「このばねは扉を静かに閉じるように働く。」 means “This spring functions to close the door quietly.” The subject in this sentence is a spring. 働く describes its function of helping the door close quietly, rather than employment at a company.
勤める: a workplace and past employment
勤める (つとめる) means to work for or to be employed at. 「姉は駅前の図書館に勤めている。」 means “My older sister works at the library near the station.” The sentence states the older sister's employment relation to the library near the station. 勤めている means she works there; it is not a statement that she visited as a library user.
「父が以前勤めた工場を見学した。」 means “I toured the factory where my father used to work.” The factory toured is where the father used to work. Read 以前 together with 勤めた to retain the past employment relation, without making him a current employee at the time of the tour.
Check the subject and the time reference
働く includes a person doing work and a thing functioning. The 勤める examples focus on a person and their workplace. Human work examples can overlap, but the spring's function should not be treated as employment.
The brother at the company, the sister at the library, and the father formerly at the factory are different subjects. Do not move all their workplaces to one person or collapse past and current employment.
How should the spring that 働く to close the door and the factory where the father 以前勤めた each be read?
Show the answer and why
The spring functions to close the door quietly; it is not employed. The factory is the father's former workplace, without a claim that he still works there. Distinguish a person's work, a thing's function, the workplace relation, and the stated time.