I wasn't going to answer as @Phoog's simple reply was enough, but then I read the comment about the Piston Counterpoint example.
Don't cherry pick examples, especially taken out of context, to rationalize mistakes in exercises.
The fig. 126 from Counterpoint you gave as an example in comments is presented in the text as: "Following are examples of such consecutive octaves and fifths." The operative word being "such", referring back to fig. 124, which is a table of 9 examples of consecutive perfect intervals in various rhythmic and melodic contexts.
Fig. 126 is given as either an example of good or bad. It's just a real example of many ways consecutive intervals can be encountered. It is an example of
(g) from the table in fig. 124. Piston's comments on that are:
...On the other hand, the formula at g seems to be open to question. It is used by Bach and others, and its acceptability appears to depend upon the rhythmic importance of the off beats.
(g) which places the octaves on the weak beats of 2 and 4 with
(f) which places the octaves on the strong beats of 1 and 3. Piston is saying weak metrical accent is a mitigating factor in the acceptability of parallel octaves. You can probably extend that to any perfect, parallel intervals.
Now that we have digested the Piston text we can return to your question.
Are parallel 5ths 8tvs allowed between beat 4 and beat 1 of new bar?
Piston's table in fig. 124 has only one example crossing a bar line with
(i). We can broaden the picture a bit and say categorically crossing the barline is a movement from a weak to a strong beat. Fig.
(e) sort of gives us that, as the first octave is on weak beat 2. But, both
(e) put the second octave not on the strong beat and instead after the strong beat. Unfortunately we don't have a perfect example. But, Piston says
(e) does not excuse the parallel movement while
(i) does, because
(i) present more rhythmic difference between the two parts.
IMO, from Piston's examples, parallel, perfect intervals crossing the barline is generally not acceptable. Certainly if the question is asked without any qualification it is not acceptable. If the idea is "crossing the bar line mitigates parallel, perfect intervals", that is certainly false. You can probably generalize that an say "any parallel movement to a strong beat will accentuate the parallel movement." And that will undermine the contrapuntal sense of melodic independence. If you do it, make sure something other than melodic contour accentuates melodic independence, the obvious choice being to make the line rhythmically independent.
Another point should be made about rhythm, meter, and these examples. In fig. 124, g the weak beats are full beat durations, while fig. 126 put the parallel octaves after both strong and weak beats with only a sixteenth note duration. Longer rhythm values are a basic way of creating metrical accent. So, teaching example
(g) accentuates the parallelism more than the real example in fig. 126. The OP question is a bit unclear on this nuance of duration.
Back to my point about cherry picking examples. If you are teaching yourself, you need more intellectual rigor when reading texts, not less. Read carefully and be a harsh critic. That doesn't mean you can't write parallel harmony any way you want. Do it, but own it. Don't pull out Piston and Bach to justify it out of context. If you want to write like Bach or Haydn, etc. be very careful about the voice leading conventions they used.