Using the greedy algorithm, this would follow if for any fixed prime q, there exist infinitely prime "pairs" of the form p and 2p-q. This follows from standard (difficult) conjectures if q is odd (for example, the case q = -1 corresponds to "Sophie Germaine Primes"). On the other hand, it would be an implication of such a partition that for each odd prime q, there either:
(i) exists at least one prime pair (p,2p-q). p,2p-q), (ii) 2q is the sum of two primes, (iii) exists at least one prime pair (p,q-2p).
Almost all proofs showing that there exist primes of a certain form also prove that there are infinitely many such primes. Thus, I suspect, one could not prove this result without also proving the difficult conjectures alluded to above.