傍ら: beside someone or alongside a main activity
Compare a cat beside a grandmother with writing novels on weekends alongside employment, distinguishing a spatial position from parallel activities in life.
A position beside a person
傍ら (かたわら) refers here to a position beside someone. “祖母の傍らには、いつも白い猫が座っていた。” says that a white cat always sat beside the speaker’s grandmother.
祖母の傍ら gives the cat’s position. 白い猫 is what was sitting, not the grandmother; the focus is their spatial proximity.
Another activity alongside one’s main work
In “彼は会社に勤める傍ら、週末に小説を書いている。”, he works for a company while writing novels on weekends. 会社に勤める identifies his employment, and 週末に小説を書いている identifies the writing activity carried on alongside it.
Keep the timing 週末に: the sentence does not require company work and writing to occur at the same moment. It describes the writing activity without stating that a novel has already been completed.
Use the preceding expression to identify the relation
After 祖母の, read a position beside a person; after 会社に勤める, read the relation between employment and another activity. The preceding expression and the rest of the sentence identify the sense of 傍ら.
Do not turn weekend writing into an action performed during work at the same instant, or into a brief opportunity taken while doing an errand. The example presents employment and writing as parallel activities in one person’s life.
Does the weekend-writing example say that he writes at the exact moment he is doing company work, or that a novel is already finished?
Show the answer and why
No. It describes writing novels on weekends alongside his employment. It does not state simultaneous actions at work or a completed novel. In the grandmother example, 傍ら instead gives a spatial position.