Electronic – Parallel caps transfer function

I am sure that they can be implemented with linear elements: resistors, inductors and capacitors.

I wouldn't be so sure about that. A trivial counter-example is \$H(s) = K, K > 1\$.

A less trivial counter-example is any \$H(s)\$ with RHP poles or any with a finite impulse response. RLC circuits cannot implement an arbitrary transfer function.

Transfer functions can be written as the product of 2nd order transfer functions (for even order) and as the product of 2nd order and 1st order transfer functions (for odd order).

Thus, as Andy aka answers, realizable transfer functions can be implemented as a cascade of well known 1st and 2nd order active filter circuits.

Now, if one wants to implement a realizable transfer function entirely with passive components, you cannot use the above method. In the case of active filters, there is no (or actually very little) interaction between the active filter sections due to very low output impedance of the op amp. However, with passive filter sections, there is interaction (loading effects) the make the choice of topology and components exceedingly complex.

The synthesis techniques of passive ladder filter networks for the canonical filter alignments is well known (see, for example) but, for arbitrary, realizable transfer functions, I don't know.

In order for the \$R\$ and \$C\$ to be in parallel, you would need \$V_{\text{out}} = 0\$ due to a short circuit. But that's not the case.

First calculate \$V_{\text{out+}}\$, the voltage at the + terminal of \$V_{\text{out}}\$. This is just a voltage divider:

$$V_{\text{out+}} = \frac{1/sC}{1/sC + R}V_{\text{in}} = \frac{1}{1 + sRC}V_{\text{in}}$$

Now calculate \$V_{\text{out-}}\$, the voltage at the - terminal of \$V_{\text{out}}\$. This is also just a voltage divider:

$$V_{\text{out-}} = \frac{R}{1/sC + R}V_{\text{in}} = \frac{sRC}{1 + sRC}V_{\text{in}}$$

Now you can calculate \$V_{\text{out}}\$:

$$V_{\text{out}} = V_{\text{out+}} - V_{\text{out-}}$$