Both my parents have European stock- ages ago yet still closely connected.

参考译文：我父母双方都有年代久远且关系仍然密切的欧洲血统/家系。

ages ago yet still closely connected为分词形容词短语作后置定语修饰名词“European stock 欧洲血统/家系”

(1)stock  n.出身;种族;家系 (英文解释If you are from a particular stock, you are descended from a particular group of people.）

We are both from working class stock.  

我们俩都出身于工人家庭。

blacks, Asians and people of Mediterranean stock.  

黑人、亚洲人和地中海沿岸高加索人的后裔

(2)connected  adj.有关系的;有联系的(英文解释：If one thing is connected with another, there is a link or relationship between them).

Have you ever had any skin problems connected with exposure to the sun?

你曾因日晒出现过皮肤问题吗？

The dispute is not directly connected to the negotiations.  

这一争端与谈判没有直接联系。

(3)closely adv.  接近地; 紧密地;密切地； 严密地; 亲近地

Working with Ford closely, I fell in love with the cinema.  

与福特的密切合作开始令我爱上了电影。

Wherever they went they were closely followed by security men.  

不管上哪儿，保安人员都紧跟在他们身后。

He had worked closely with the President during his governorship.  

他在任州长期间与总统紧密合作。