As described in Srimad Ramayana, and is known to all, Maharaja Dasaratha, the father of Sri Rama, was emotionally attached to his son - so much that he wasn't able to bear separation from his son for a few days with Brahmarshi Vishvamitra, and later (eventual) 14 years.
But earlier in the Ramayana, Dasaratha tells Vishvamitra:
Sixty thousand years have passed from my birth, oh! Vishvamitra, and this Rama is engendered at this age, that too with tribulations, hence taking Rama with you will be inappropriate of you. [Balakanda, 20th Sarga, 10b-11a]
So to a person who has lived for 60,000 years, surely 14 years, and a few days in Vishvamitra's case cannot be of much significance? One can justify the latter in stating that Rama had just returned from Gurukula, and that Vishvamitra was taking him to fight demons.
Furthermore, Dasaratha is perfectly okay when Yudhajit, Kaikeyi's brother takes away Bharata and Shatrughna for 12 years after their marriage, to Kekaya. So when he is not disturbed by their long time away from home, why is he so distraught at Rama's exile that it forces him to give up even his own life?