If you have a Cook reduction in both directions, do you also have a Karp reduction?
The answer depends on whether you require both of $\mathcal{P}_1,\mathcal{P}_2$ to be in NP or not.

If the problems need not be in NP

Then the answer to your question is "No".  There exist problems $\mathcal{P}_1,\mathcal{P}_2$ that are Cook-reducible (in both directions) but not Karp-reducible (in either direction)

In particular, consider $\mathcal{P}_1 = $ HALT (the Halting problem) and $\mathcal{P}_2 = \overline{\mathcal{P}_1}$ (its complement).  They are Cook-reducible (you just invoke the oracle and complement its answer), but not Karp-reducible.

(Similarly, if $\textsf{NP} \ne \textsf{coNP}$, then the answer is No: take $\mathcal{P}_1 = $ SAT and $\mathcal{P}_2 = $ TAUTOLOGY, so that $\mathcal{P}_1$ is NP-complete and $\mathcal{P}_2$ is coNP-complete.  They're Cook-reducible, but a Karp reduction would prove that $\textsf{NP} = \textsf{coNP}$ via a standard argument.)

If the question is restricted to require both problems to be in NP

Then it's an open question; no one knows whether the answer is "Yes" or "No".  See https://en.wikipedia.org/wiki/NP-completeness#Completeness_under_different_types_of_reduction and 


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https://cstheory.stackexchange.com/q/138/5038 and
https://cstheory.stackexchange.com/q/3435/5038 and
https://cstheory.stackexchange.com/q/14379/5038 and
https://cstheory.stackexchange.com/q/16983/5038.