As Nagora mentioned in his answer, this is a grey area that will require DM interpretation. Here are a few possible approaches to this problem:
- A single save vs. fear that lasts the entire encounter
- A save vs. fear whenever the creature does specific actions
- A save vs. fear each round as long as the encounter lasts
- A save vs. fear once every set number of rounds
Now any of those options is defensible -- the first lessens the workload of running the encounter, the second seems to fit the RAW text best, while the last two make these creatures extra fearsome -- so you should feel free to do what works best for your group and campaign. Just be sure to inform your players ahead of time if you're making a change (i.e., before they commit to a strategy based on your previous ruling) and enforce your choice consistently going forward.
Addendum about dragon fear in the RAW: The best evidence from the core 1e books we have to adjudicate this comes from the MM on pages 29-30, which says the following about dragons:
"At adult age and older they radiate a powerful aura which causes a fear reaction, when a dragon flies overhead or charges, as follows ... [specific rules for different hit dice characters/creatures]"
This suggests that the fear reaction occurs each time the dragon does one of two things: charges the characters or flies over them. If running the rules as close to the RAW as possible matters to you, I would suggest the second method from the above list.