Mr Bates, 43, and Fraser, seven, were struck on the A4119 at Talbot Green at about 00:30 GMT on Sunday.
A 22-year-old driver of an Alfa Romeo has been arrested and bailed, South Wales Police said.
Wife and mother Anna-Louise Bates called Mr Bates "my wonderful husband" and Fraser "our bear".
The pair were struck while crossing the carriageway near the Arthur Llewellyn Jenkins furniture store, near Llantrisant.
Mr Bates, from Llanishen, Cardiff, an IT programme manager for PHS in Caerphilly, died at the Royal Glamorgan Hospital shortly after the crash.
Fraser, a pupil at Llysfaen Primary School in Cardiff, died on Sunday night at the Bristol Children's Hospital.
Mrs Bates said: "Stu was my Mr Wonderful. Fraser was 'our Bear' and his initials spelled out FAB - he was indeed Fab by name and Fab by nature.
"Together with three-year old-Elizabeth we were a family of four, but I know everybody we met over the years became part of our extended family."
Mrs Bates said she believed the pair would be singing the song 'Forever Young' in heaven while playing football with Fraser's hero, former Manchester United footballer Duncan Edwards.
"Look after each other my bears," she added.
Headmistress of Llysfaen Primary School, Melanie Sancto, described Fraser as "a bottle of pop, a fun-loving, beautiful little boy".
She said: "He had lots of friends and was so well thought of by everyone here.
"We will all remember him as happy, funny and, most of all, football mad. The game was a huge part of Fraser's and Stuart's lives."
Ms Sancto added that Mr Bates played a big part in the school community.
"Our heartfelt thoughts, support and love go out to Anna-Louise and the family," she added.
A funeral will take place at St Martin's Church in Roath, Cardiff, at a later date.
South Wales Police is continuing to investigate the crash.
A father and son who died after being hit by a car in Rhondda Cynon Taff have been named by police as Stuart and Fraser Bates.