In the name of Allah the most compassionate the most merciful
First of all, it can be helpful for you to be aware that:
After the martyrdom of Hazrat Fatimah-a-Zahra, as the wife of Imam Ali
(a.s.), Imam Ali sent his brother (Aqil ibn Abi Talib who was familiar
with to Arab Ensaab (genealogy)) and asked him to find him a wife from
a valiant and brave family till …)
Aqil (a.s.) chose Um-al-Banin-Kalabiah for Imam Ali, that her tribe and family was incomparable and unexampled in bravery. In truth, Bani-Kalab were proverbial and famous for their bravery between Arabs. Moreover it is narrated that her mother was Thomamah as a true-hearted and virtuous (piety) woman, therefore understandably it can be inferred that both of her parents would be considered as two positive persons who were at the way or followers or Sahabah of the Prophet (and Imam Ali as you asked), otherwise logically Imam Ali wouldn’t choose them. For more info. as helpful info. regarding what you need, I assume the following book would be related for you. مروج الذهب، المسعودي، ج۳، ص۶۳./ العقد الفرید، احمد بن عبد ربه، ج۵، ص۱۳۴.
In the meantime, as a helpful matter regarding "if her father was from Sahalba" @ Medi added in his comment:
her father (May Allah be pleased with him) is among the sahaba this is
quoted in al isaabah from ibn Hajar al 'asqalani! – Medi Saif
In regard to the Nasab (descent) of Um-al-Banin (s.a.) (acnd actually the descent of his father, Hazam ibn Khalid), you’d better see the following issue which God willing would be constructive and helpful for you about her father and mother.
At the initial step, in regard to her father, she was Fatimah, the daughter of:
حزام بن خالد بن ربيعه بن وحيد بن كعب بن عامر بن كلاب بن ربيعة بن عامر
بن صعصعة بن معاوية بن بكر بن هوازن
And her mother was from the family of:
سهل بن عامر بن مالك بن جعفر بن كلاب
And her Jaddah or ancestress include:
جده اول : عمره دخت (بنت) طفيل بن مالك احزام بن جعفر كلاب .
كبته دخت (بنت) عروة الرحال فرزند جعفر بن كلاب
3th: ام خشف دخت (بنت) ابى معاوية فارس هزار بن عبادة بن عقيل بن كلاب
4th: فاطمه دخت (بنت) جعفر بن كلاب
5th: عاتكه دخت (بنت) عبدالشمس بن عبد مناف بن قصى ، جده حضرت رسول صلى
الله عليه و آله
آمنه دخت (بنت) وهب بن عمير بن نصر بن قعين بن حرث بن ثلعبه بن ذودان بن
اسد بن خزيمه
7th: دخت (بنت) جحدر بن ضبيعه اغر بن قيس بن ثلعبية بن عكاية بن صعب بن
على بن بكر بن وائل بن ربيعه بن نزار
جد حضرت رسول خدا صلى الله عليه و آله
8th: دخت (بنت) ملك بن قيس بن ثعلبه
دخت (بنت) الراءسين : خشين بن ابى عصم بن سمح بن فزاره
دخت (بنت) عمر بن صرمة بن عوف بن سعد بن ذبيان بن بغيض بن ريث بن غطفان .