In The Eye of the World, when Tam is wounded, Rand has not yet started channeling, at least not in any viable way. Washing away Bela's fatigue was very simple and required little power compared to actually healing someone with a serious wound and infection and it didn't happen until after he had left Emond's field. In fact that was probably the first time that he actually channeled.
"It is during this journey that hints are given that Rand can channel; when he gets goose bumps on his arms when Moiraine channels near him and when the group are healed from their fatigue but Bela is perfect as Rand already healed the horse (for which he suffers a bout of reckless bravado with a trio of Whitecloaks). His first big use of saidin is when the Darkfriend Howal Gode tries to attack Rand and Mat. Rand blasts a hole in the wall, blinding Mat temporarily. He then suffers flu-like symptoms after this heavy use of the One Power."
Even Nynaeve, who was an accomplished healer in her own right and had been unknowingly using the One Power for some time as a wilder could not heal Tam.
"She had been the youngest Wisdom the Two Rivers had ever had, and one of the most successful. She was said to be able to cure ailments no other Wisdom ever could and wounds that would leave a man maimed for life were sometimes gone without a scar [verify]. She was the only Wisdom in the Two Rivers within the span of the books who can truly Listen to the Wind."
So it stands to reason that at that time he did not have the power to heal his father no matter how worried or desperate he was and only later was able to perform great feats without understanding how.
I hope this answers your question. Link to my info source below. Tai'shar Malkier!