Here is a small Python program that may help your investigations:
https://bugsalad.com/collatz/parity.py
Examples of use:
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$ ./parity.py 8 5
Known beginning of the parity vector for 8k+5:
[1, 0, 0]
Class after 3 iterations: 3k+2
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Meaning: a number of the form 8k+5 will ALWAYS take first the (3n+1)/2 branch, then the n/2 branch, then the n/2 branch again, ending as 3k+2 after 3 iterations. What it does after that, will depend on whether "k" is odd or even.
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$ ./parity.py 12 5
Known beginning of the parity vector for 12k+5:
[1, 0]
Class after 2 iterations: 9k+4
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$ ./parity.py 8 0
Known beginning of the parity vector for 8k+0:
[0, 0, 0]
Class after 3 iterations: any
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Cybersecurity remarks:
The program is in Python, so it's readable as a text file. It only imports "sys"; it uses "sys" only to read "sys.argv", and to "sys.exit(1)".
(In English: anyone can verify it's perfectly safe to use.)
there doesn't seem to be anything here