Do monks not have responsibility towards parents and family? In this age, getting enlightened is not sought after by many as in Buddhas era. So I assume it must be harder for parents to accept such a decision. But monks who decide to ordain also have responsibility towards parents. How does that fit into ethics of Buddhas teachings? 

I understand giving Dhamma to parents is one of the greatest gift one can give to a parent. Can it not be done by living with them and still maintaining lay practice (like the potter who was an anagami and looked after his parents) Why ordain? 

NB: this is asked with due respect and do not wish to condone monks or people who desire to ordain. As Buddha would not have been a Buddha if he did not step out. I just want to understand ordaining in today's day and age.
Responsibility in Theravada Buddhism falls into two categories - practical and absolute. In an absolute sense, we only have two responsibilities:

“bhante, imasmiṃ sāsane kati dhurānī”ti?
&quot;Bhante, in this religion, how many duties are there?&quot;
“ganthadhuraṃ, vipassanādhuranti dveyeva dhurāni bhikkhū”ti.
&quot;The duty of study and the duty of insight. These are the only two duties, bhikkhu.&quot;
-- Dhp-A 1

In a practical sense, a lay person has duties towards six groups of people:

&quot;And how, young householder, does a noble disciple cover the six quarters?
&quot;The following should be looked upon as the six quarters. The parents should be looked upon as the East, teachers as the South, wife and children as the West, friends and associates as the North, servants and employees as the Nadir, ascetics and brahmans as the Zenith.
-- DN 31 (Narada, trans)

These are considered practical responsibilities, because obvious practical problems arise when a lay person fails to fulfil them.
Generally speaking, a monk need concern him or herself with practical responsibilities towards fellow monastics (including teachers and preceptors) and lay disciples.
That being said, a monk may at times find him or herself in a position of needing to fulfil practical responsibilities to one's parents; if they are sick, destitute, etc. To this end, there are disciplinary exceptions across the board for taking care of one's parents, including in regards to giving medical treatment and food, visiting during the rains retreat, etc.
This would seem to suggest an obviously practical encouragement for monks to care for their parents as appropriate / necessary.