It wouldn't be hard for an accident to happen. Imagine just both the Parliamentarians, Lords and French soldiers shouting at each other and in the confusion a single soldier fires his musket on accident, the other french soldiers follow suit believing their office gave the order to fire (similar to the Boston Massacre). The Parliamentarians attack the soldiers and the soldiers fight back, killing most or all.
That´s a bit impractical. The Parliamentarians would not fight. They would run, or surrender individually. Any fighting would be done by the soldiers defending Westminster Palace
The French would not be aiming to kill the whole Parliament en masse. Even if they ARE going to kill everyone they get, they are not going to do it then and there. The Parlamentarians they do kill will be kept alive a few months, subjected to show trials for high treason against James III and VIII, and hanged drawn and quartered at Tyburn, or hanged till dead or beheaded if the King wants to display mercy.
How many pardons James III was planning for in 1759 was another matter. Nearly every single politicians alive in 1688 was dead long ago - exhumation was an option (as with Cromwell, Bradshaw and Ireton), but Charles in 1660 had exhumed and executed just 3 out of 20 or so dead commissioners (and at least 2 exhumed were reburied without execution), and hanged drawn and quartered 6 out of 30 or so taken alive (the rest got the mercy of life imprisonment or better).
Anywaya logical scenario that might lead to mass fatalities of Parliament might be a botched defence of Westminster.
Westminster was unfortified as of 1759. The Lines of Communication had been demolished in 1647. The gates of City walls had been unhinged and portcullises wedged open in 1660, though they would be demolished... around 1760. Westminster Bridge was completed in 1750.
The Parliament does not plan on dying in defence of their seats. If they expect Parliament to be less safe than adjourning to flee individually overland and by boats then they will scatter to flee.
But assume that a cavalry raid does suddenly show up in the streets of Westminster, and before the Parliament can scatter, a few hundred horsemen block Westminster Bridge, Strand and perhaps the Thames embankment (again perhaps with unfavourable tide to leave by boat before the embankment is blocked). If the Parliament expects it to be a probing cavalry raid, and hundreds of infantry to be arriving from City every minute, will they surrender (inside or outside the house), or try to stay safe for these few minutes under the protection of a few tens of their infantry holding the building entrances?
And what if the Houses, while attempting to await rescue that does not come or comes too late, catch fire? Well, they are not storing gunpowder in Westminster in 1759. I expect that the palace had enough exits, and that they would attempt to surrender or try their luck for breakout rather than burn.
I'm not sure about the King and Royal family fleeing, but the closest I could refer to is the Portugese Royal Family and court moving to Brazil in late 1807. So it wouldn't be a far stretch to suggest the British Royal Family could do the same in 1759.
It is NOT British Royal Family in 1759. It is Hanoverian Royal Family.
So, they are on the run. Where?
Hanover is nearby. But who do they run with?
What are the rank and file British sailors, or officers, going to do with their ships if they are expecting an unsympathetic Jacobite restoration on the point of French bayonets?
Many will want to give up anyway and go home. How many are willing to sail for indefinite or lifetime exile? Well, Ireland in 1691 had 14 000 wild geese.
There are navigational constraints as to where the navy can go. They need ports with dockyard infrastructure for maintenance, or they will be unseaworthy rotting hulks soon.
Which ports did Hanover have, in 1759, for sea travel to England? Hanover certainly did not have a large navy. The Navy on the run in 1759 might disembark passengers in Hanover and take water and food, but it cannot remain a navy there.
If the sailors go to land in Hanover, they will spend the rest of their lives as Englishmen in a poor German speaking country where they are resented and at best pitied, but loved by nobody.
The other options for the navy are either, in Europe, allied countries (who in 1759?), which means being bumped again when the allies make separate peace - or sailing for English speaking Americas, where they can also found natural harbours and plenty of naval timbers.
Now, where will George II go? And will his chosen direction affect the decision of the navy? Is the navy willing to go to America to fight for King of Hanover who flees to Hanover? Will it make a difference for them if their King shares their exile?
(Hanoverians are used to King beyond the water.)
Also as a matter of curosity, if George III comes to the throne such as he does in OTL who would he marry in North America? The daughter of a colonial plantation owner or would he have a european noble brought over?
European royalty. No plantation owner will have any influence to be useful.
If France of 1759 succeeds in overthrowing Great Britain and imposing James III on swordpoint, expect balance of power hostility to overmighty France. The King-of-England-in-exile will be desperate for a political alliance in Europe. If they do get one... would George III settle for a Catholic Habsburg if no one else dare stand up to France? Or would they pick one of equal-birth, Protestant minor German princes dispossessed by French as ex-Hanover allies? The major Protestant rulers being Orange stadholder of United Provinces, and kingdoms of Prussia, Denmark and Sweden.