Percentage increase with parallel pipe at midpoint

Setting up. Let the single pipe have length $L$, diameter $D$, friction factor $f$, joining two reservoirs with a level difference $H$. Minor losses are neglected, so the head is lost in friction only. Use $h_f = \dfrac{fLV^2}{2gD}$.

Before

$$H = \frac{fLV_1^2}{2gD} \;\Rightarrow\; V_1 = \sqrt{\frac{2gDH}{fL}}, \qquad Q_1 = AV_1$$

After

From the mid-point, a second pipe of the same diameter and length $L/2$ runs parallel to the second half. The first half carries all the flow ($V$); the second half has two identical pipes, so the flow divides equally and each carries velocity $V/2$.

$$H = \frac{f(L/2)V^2}{2gD} + \frac{f(L/2)(V/2)^2}{2gD} = \frac{fLV^2}{2gD}\left(\frac12 + \frac18\right) = \frac{fLV^2}{2gD}\left(\frac58\right)$$ $$V_2 = \sqrt{\frac{8}{5}}\sqrt{\frac{2gDH}{fL}} = \sqrt{1.6}\,V_1 = 1.2649\,V_1$$

The discharge in the first half is $Q = AV$, so

$$\frac{Q_2}{Q_1} = \sqrt{1.6} = 1.2649$$ $$\boxed{\text{Increase in flow} = 26.5\%}$$

(This value does not depend on $D$, $L$, $f$ or $H$.)