What you've tested using that invocation is the
    null hypothesis that the underlying probability
    of 39/67 is 0.5.

Isn't that equivalent to my interpretation of the test result? "In a world where it makes no difference which design is used, you would get a result as significant as this 22% of the time".

    If you want to perform a test of a difference of two proportions, you need to do:
    prop.test(c(39, 67), c(total_group_a_impressions, total_group_b_impressions))

Do you mean c(39,28)? Because group_a had 39 hits and group_b hat 28. Doing so with the group sizes Bemmu stated (3000/3000) also gives me a p value of 0.22.

As long as the group sizes are equal, the test is not very sensitive to the sizes.