First of all, your observation is correct. Second, this is by far not the only path construction that does not have a "timer". For instance, this is also true for the sin and cos path constructions, and of course plots.
\documentclass[margin=5pt, tikz]{standalone}
\usepackage{pifont}
\usepackage{amssymb}
\begin{document}
\begin{tikzpicture}[]
\draw[] (0,0) -- (3,5) coordinate[pos=0.675, label=N](N)
node[circle,draw]{\ding{51}};
\draw[blue] (0,0) arc (0:75:4) coordinate[pos=0.675, label=A](A) node[circle,draw]{\ding{51}};
\draw[red, shift={(2,0)}](0,0) parabola [parabola height =3cm](4,0)
coordinate[pos=0.675, label=P] (P) node[circle,draw]{\ding{55}};
\draw[cyan, shift={(6,2)}](0,0) sin(2,1)
coordinate[pos=0.675, label=Q] (Q) node[circle,draw]{\ding{55}};
\draw[cyan, shift={(6,4.5)}](0,0) cos(2,-1)
coordinate[pos=0.675, label=R] (R) node[circle,draw]{\ding{55}};
\end{tikzpicture}
\end{document}
In fact, the pos syntax only really works for
- straight lines (
\tikz@timer@line),
- single Bezier curve stretches (
\tikz@timer@curve), and
- arcs (
\tikz@timer@arc).
So what is going on here? Internally TikZ works with a "timer", the relevant macros from tikz.code.tex are \tikz@timer (as well as \tikz@timer@start and \tikz@timer@end). As far as I can see, only \tikz@timer@line (with the subcases \tikz@timer@hvline and \tikz@timer@vhline), \tikz@timer@curve and \tikz@timer@arc are implemented. If your question is whether or not one could add a timer for parabolae, sines and cosines, the answer is likely affirmative, but given the complexity of the other timers in tikz.code.tex the actual realization will be messy.
So, in order to answer the question
How to place a coordinate at parabola-path-position?
I recommend looking at the decorations.markings library.
\documentclass[margin=5pt, tikz]{standalone}
\usetikzlibrary{decorations.markings}
\begin{document}
\begin{tikzpicture}
\draw[red,postaction={decorate,
decoration={markings,
mark=at position 0.675 with {\coordinate[label=P](P);}}}]
(0,0) parabola [parabola height =3cm](4,0) node[below]{works};
\end{tikzpicture}
\end{document}
(If you encounter dimension too large errors with this decoration, some of them can be cured with use fpu reciprocal that ships with circuitikz.)
As a side remark, if you use pgfplots the pos key does work for one-dimensional plots, even for pretty complex ones.
