Imho unlikely unless the bike was a bitsa.
First photo - View attachment 53198
- depends what the shiny part between your father's leg and the cylinder head is; if it is a carburettor and not in the centre of the bike, it is one of two, makes the bike a Bonneville but then the photo is not late 1950's - the first Bonnevilles were 59 and had the signature nacelle enclosing the headlamp and fork tops, mounting the speedometer, Ammeter and light switch. Bonnevilles did not come without the nacelle until 60, 650 engines became unit from 63.
If the shiny part is a single carb, tank badge is 57 on. 58 Speed Twin was the last pre unit (crankcase and gearbox separate), 59-66 Speed Twins have unit engines. However, Speed Twins had the nacelle and the 58 Speed Twin had deeply valanced fenders, the bike in the photos does not have either the nacelle or the 58 fenders.
Also unlikely is 57, 58 or 59 Tiger 100 (the Tiger 100 did not get the unit engine until a year after the Speed Twin) just because all the online photos I can find show they had the nacelle too.
If the photo is definitely late 1950's and the bike was not a bitsa built by your father, I think it might have been one of the official Triumph US market 650 bitsas, originally nicknamed the "Trophybird"
Triumph had built the off-road "Trophy" version - first 500, later also 650 - since winning the 1948 International Six Days Trial, with the headlamp enclosed in a separate chromed shell (so it could be detached easily for racing) and the fenders in the photos. Otoh, the on-road 650 - Thunderbird and Tiger 110 - were also supplied with the nacelle and deeply valanced fenders, much disliked in the US. Not sure which dealer built the first but soon several were stripping nacelles and fenders off Thunderbirds and Tiger 110, replacing them with Trophy parts. Meriden eventually gave into pressure from the US importers and, along with the 60 onwards Bonneville, built both on- and off-road versions of the Trophy without nacelle or valanced fenders.
Do you remember the colour of your father's bike? Speed Twin (500) cycle parts were painted a dark red ("Amaranth Red") usually including the frame, swinging arm and rear brake plate. Otoh, Thunderbird (650) cycle parts were painted blue, Tigers and Trophys (500 and 650) were usually black and silver/chrome.