Why did Elrond let Pippin be a part of the Company of the Ring?

Question

Frodo's three hobbit companions in The Lord of the Rings were not yet "of age" (33) when they left to go to Rivendell with no permission from their parents.

Knowing their age, why did Elrond finally allow these hobbits to go with the Company? Is there an indication that hobbits get freedom before they come of age? In RotK, Pippin says:

"I am still little more than a boy in the reckoning of my people, and it will be four years before I 'come of age', as we say in the Shire."

Correction:

Sam and Merry are of age.

Answer 1

Corrections

Your question makes a questionable assumption:

Frodo's three hobbit companions in The Lord of the Rings were not yet "of age" (33) when they left to go to Rivendell

This is not entirely true. Pippin was not yet 33, but Merry and Sam both were; the birth years of each of the four Hobbits is given in Appendix C as (years in Shire Reckoning):

• Sam: 1380
• Merry: 1382
• Pippin: 1390

The events of The Lord of the Rings occur in the year S.R. 1419, meaning Sam was 39, Merry was 37, and Pippin was 29.

So I'm going to modify your question slightly:

Why was Pippin allowed to join the fellowship, despite not being of age?

Elrond actually does raise doubts about Pippin, specifically because of his age; he relents largely because Pippin makes it clear that he intends to go anyway, and Elrond clearly doesn't want to imprison him:

In any case, I judge that the younger of these two, Peregrin Took, should remain. My heart is against his going.'

'Then, Master Elrond, you will have to lock me in prison, or send me home tied in a sack,' said Pippin. 'For otherwise I shall follow the Company.'

'Let it be so then. You shall go,' said Elrond, and he sighed. 'Now the tale of Nine is filled. In seven days the Company must depart.'

^{Fellowship of the Ring Book II Chapter 3: "The Ring Goes South"}

Answer 2

Elrond saw the devotion that Sam had for Frodo and realized it would be better to send him along for support.

"But you won't send him off alone surely, Master? cried Sam, unable to contain himself....

No, indeed! You at least shall go with him. It is hardly possible to separate you from him, even when he is summoned to a secret council and you are not'

• Chapter 2, FotR

Gandalf convinced Elrond to send the others.

It is true that if these hobbits understood the danger, they would not dare to go. But they would still wish that they dared, and be shamed and unhappy. I think, Elrond, in this matter it would be well to trust to their friendship than to great wisdom.

• Chapter 3, FotR

Additionally, - they were at risk because of what they already know of the Ring, so better to send them along with Gandalf instead of back to the Shire unprotected.