Does there exist a problem for which no existing solution can be parallelized on a Turing machine?
The question is not precisely defined, so it's not really amenable to a precise technical answer at this stage.  But a common view/hypothesis/expectation is that there exist problems that are inherently sequential, in the sense that they cannot be substantially sped up on a parallel computer, no matter how many parallel processors you have.  A weak version of this view is implied by the conjecture that $NC \ne P$ (see https://en.wikipedia.org/wiki/NC_%28complexity%29), which is conjectured but not known to hold.  In particular, P-complete problems are good candidates for problems that don't benefit much from parallelization.  There are stronger versions of this view; for instance, in cryptography, timelock puzzles utilize functions that are believed to be inherently sequential: a parallel processor cannot speed them up very much.