Is there a generic way to parallelize algorithms?
No, there cannot be such a generic method to produce algorithms $A_{in}, A_{out}, A_{1},\dots, A_k$ with the requested properties. The basic problem is that the requested parallel algorithm only uses a single round. (And allowing $A_{in}$ and $A_{out}$ only $O(k)$ time is very restrictive, they cannot even read length-$n$ input or write a length-$n$ output in that time.)

One simple problem which even has a parallel algorithm (but not of the form stated in the question) is the all-prefix-sums operation. Input is a list of numbers, and output is another list, whose elements are the sum of all previous elements in the original list. See section "3.2 Scans" on page 23 in

Parallel Algorithms (64 pages)
https://www.cs.cmu.edu/~guyb/papers/BM04.pdf
by Guy E. Blelloch and Bruce M. Maggs – 1996



There are at least two possible criticisms of this answer. (1) The counter-example problem took a list of size $n$ and output a list of size $n$. Maybe the question expected the output to be much shorter than the input, like in a decision problem. (2) No explicit proof is given that the counter-example problem has no parallel algorithm of the form stated in the question.

Regarding (1), an output of length-$n$ is used to get around Barrington's theorem, as explained in a comment below the question. Regarding (2), one might look at the elements from $n/2$ to $n$ of the output. Each of them would require more than $n/2$ additions, if computed in isolation. $A_{out}$ can only output $O(k)$ of them in $O(k)$ time, so most of them must be computed in partial isolation. Hence, the algorithms $A_i$ outputting them will need more than $O(n/2)$ time.