The regula falsi algorithm is based on a linear interpolation between the points $a$ and $b$, which bracket a root we want to find. Would it be any improvement to use a parabolic interpolation instead?
From an algorithmic point of view, it seems to me the modifications needed are minimal, it wouldn't even be needed to store additional data. It suffices to compute the new approximation just after we have all three $f(a)$, $f(b)$ and $f(c)$ values, before replacing $a$ or $b$ with $c$. Fitting the three points to a parabola is relatively simple, and even getting the root between $a$ and $b$ seems easy.
I didn't see such modification being suggested anywhere. There's Muller's method, but that's secant-like rather than regula falsi-like. Another method that looks similar is Ridders' method, but that uses an exponential interpolation and requires an additional evaluation at the interval midpoint (is it possible to use Ridders' method with an arbitrary $c$ point between $a$ and $b$?).
Let me answer at least partly your question.
Suppose that you have three points $f(a), f(b), f(c)$. For sure, you could compute the equation of the parabola. But, you want $f(x)=0$; this then implies that yuou solve the quadratic and select the right root of it (I a m sure we could find symptomatic situations in which both roots could be in the interval). This is why Muller proposed an inverse quadratic interpolation ($x$ as a function of $y$) since you just need to apply $y=0$ to get the next iterate. Ridder's method is basically in the same spirit (no equation to solve).
By the way, I found an interesting paper for you :
http://www.cscjournals.org/csc/manuscript/Journals/IJEA/volume4/Issue1/IJEA-33.pdf