Independent or common draws when sampling different posteriors

Suppose you have 30 posteriors for each of two parameters p and q, all fitted in the same model [p1,…,p30] and [q1,…,q30].

To test for a relation between p and q, you sample b times from posteriors, in each case computing a Pearson coefficient between vectors [p1(k),…,p30(k)] and [q1(k),…,q30(k)] where k is the element drawn from posteriors on each bth sample.

Should the sampling be done using the same index on each draw, i.e.,
[p1(k),…,p30(k)] and [q1(k),…,q30(k)]

or a different index for the different posteriors, i.e.,
[p1(k),…,p30(k)] and [q1(i),…,q30(i)]
where k and i are generated independently

or a different index for each posterior entirely
[p1(k1),…,p30(k30)] and [q1(k31),…,q30(k60)]
where each index [k1,…,k60] is generated independently