Why to do parentheses change the results of a calculation?

Question

I'm getting results that are sensitive to where I place parentheses with respect to operations that are associative¹ (and should thus be insensitive to such placement). For example, if I define²

<< Units`; << PhysicalConstants`;
stellarDayTextbook = 1/(1/Day + 1/Convert[SiderealYear, Day])

and then calculate

1/stellarDayTextbook - 1/Day - 1/Convert[SiderealYear, Day]

I get precisely zero, as expected. But if I add parentheses

(1/stellarDayTextbook - 1/Day) - 1/Convert[SiderealYear, Day]

I get

-(3.03577*10^-17/Day)

What's causing this?

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^{1. Remember, this isn't C, it's math: "subtraction" of $x$ is just the addition of $-x$. Check TreeForm.}

^{2. I realize this isn't the definition of a "stellar day", but merely a textbook approximation. The distinction is not material to the question.}