I think it might have been:

the Cook

My train of thought:

After I posted my original answer, I've noticed that the question has been tagged with logical-deduction and liars. I could quickly explain the logical deduction I used with the updated answer. As for the liars, I failed to come up with a story where the liars' lie would mean something significant to the case, until now.

Liars:

Father and Cook

Reasoning:

Father drove both the Sister and the Eldest Brother away from home, because he had to work from home.

Cook joined Father in the study in the afternoon, expecting no one else to be at home. She "glimpsed the eldest boy through the crack in the door while in there."

I can't personally recognize anyone passing through the hallway from a crack in the door. But I can certainly at least hear which door has just been opened/closed on the same floor.

So I bet the study was not where the only place they stayed. A logical guess would be the Parent's Bedroom. And they might have been seen by the eldest boy instead of the other way around.

Motive:

the eldest brother blackmailed the pair

Method:

food poisoning

• might update answer later with more details

I think it might have been:

the Cook

My train of thought:

After I posted my original answer, I've noticed that the question has been tagged with logical-deduction and liars. I could quickly explain the logical deduction I used with the updated answer. As for the liars, I failed to come up with a story where the liars' lie would mean something significant to the case, until now.

Liars:

Father and Cook

Reasoning:

Father drove both the Sister and the Eldest Brother away from home, because he had to work from home.

Cook joined Father in the study in the afternoon, expecting no one else to be at home. She "glimpsed the eldest boy through the crack in the door while in there."

I can't personally recognize anyone passing through the hallway from a crack in the door. But I can certainly at least hear which door has just been opened/closed on the same floor.

So I bet the study was not where the only place they stayed. A logical guess would be the Parent's Bedroom. That's where the Cook saw the eldest son. And they might have been seen by the eldest boy as well.

Motive:

the eldest brother blackmailed the pair for a promotion

hence, the sudden argument between Father and Grandfather

Method:

food poisoning

I just don't see how either of them know about the drug the Eldest Brother had been using though, so it might not have been drug overdose in this case.