Direct train Latour de Carol to Perpignan

There isn't. You need to change in Villefranche - Vernet-les-Bains and currently also in Ille-sur-Têt, because there's a temporary bus replacement. Timetables on www.sncf-connect.com.

Hi,
The journey is 2 trains - The Train Jaune and then a conventional TER LIo regional train - both bookable (and current time tables) on www.tersncf.com/occitanie. Currently ( May 2025), the Villefranche-Perpignan line has (as others have said) a replacement bus from Villefranche to Ille-sur-Tet due to a landslide causing a train to demolish a bridge.  One can also take the TER LIo bus #560 from Villefranche cite (car park by the gates at the station end) to Perpignan combined train and bus station for 1 euro (with a possibly friendlier schedule).
There seems to be a lot of confusion regarding the schedules and booking arrangements for the Train Jaune.  As I live and run a small Chambres d’Hotes (BnB) at Villefranche-de-Conflent (one end of the TJ line - the station is called Villefranche/Vernet-les-Bains but is actually in Villefranche) I thought I’d add some more comprehensive info:
Schedules and information about temporary changes, as well as bookings can be found at www.ter.sncf.com/occitanie.  If you use an in-browser translation tool - turn it off to make bookings - it may stop the process working.  This website is the most direct way to book as it cuts out the often erroneous third party sites.
Notes about booking:
1) The TJ seat bookings are often only opened for booking about 3 weeks before they run, so don’t expect to be able to book 6 months before (but remember, neither can anyone else!).
2) Schedules change twice a year, and the timetables are usually issued for mid December to mid May and Mid May to Mid December.
3) If trains show as not available shortly before departure, they may well be full.
4) The first weekend of every month the TJ runs at a discounted price of €5 each way, so demand on those weekends can be high.
5) There are usually a handful of weeks each year in autumn and spring when the line is closed for essential maintenance.  These will be shown in the ‘current timetables’ section of the SNCF website with revised timetables.
6) When running there are multiple shorter trains during the day plying between Villefranche and Font Romeu / Mont Louis.  
7) Full distance trains between Latour-de-Carol/Enveitg  and Villefranche/VLB run thus (times change slightly through the seasons): 
Around 8AM: Latour de Carol>Villefranche
Around 10AM: Villefranche>Latour de Carol (Arr around lunchtime)
Around 15.00 (3PM): Latour de Carol> Villefranche (Your return train after a nice meal at the Bistro de la Gare!) Arrives in Villefranche around 18.30 (6PM) 
Around 16.30 (4.30PM): Last train of the day Villefranche>Latour de Carol (Arr around 19.00 / 7PM)
8) The first and last train interface nicely with the Paris-Toulouse-Latour night sleeper trains.
There are many more English language resources about the Train Jaune and Villefranche-de-Conflent itself at our website: www.chambreschouettes.fr 
I hope that helps!