Is there a limit to the number of times that a bill can be bounced between the Commons and the Lords before the Commons are allowed to use the Parliament Act?
According to Parliament UK the Lords can delay a bill for a year, not a number of times of being bounced between the houses.
Parliament Act 1949 The Parliament Act 1949 further reduced the Lords' delaying powers to one year.
This is probably related to the rules that prevent the House of Commons from re-introducing a unchanged bill in the same session of parliament
When a bill has been rejected, or lost through disagreement, it should not, according to the practice of Parliament, be reintroduced in the same session
The link between sessions and years further muddies the water here. The Parliaments page contains a list of Bills that the Parliament Act was used on, and a number where it was intended to be used;
In addition, three Bills have been introduced in a second session with a view to invoking the Parliament Act procedure but all were eventually agreed by the Lords in the second session:
Again, this references a session. Generally sessions of parliament in the UK last a year from Queen's speech to Queen's Speech. Recently however this has not been the case. With one session of Parliament running from 2017 to 2019, followed by a very short session between Johnson's prorogation and the 2019 GE.