I was thinking about the question "How old is your father?". I know that I can use oikutsu for "How old are YOU." But can I use it also for "she / he"? So, "how old is your father", could I ask "あなたのお父さんはおいくつですか。" ? And yes, I use "anata", you can also think just of a name! :) Thanks!
Short answer: Yes, you can.
Long answer: Yes, you can, but you must be careful not to make an easy-to-make mistake that could make it rude.
「幾つ（いくつ）」 originally comes from the Japanese way of counting (ひとつ、ふたつ）, with 「幾」 here meaning "how many".「何歳（なんさい）」 is obviously the 音読み way of reading derived from the old Chinese pronunciation.
When you ask a child how old they are, you ask 「いくつですか？」. However, when you ask an adult, it would be extremely rude to ask them the same way you would a child. Make sure to add the 「お」 as you have already correctly done to avoid offending anyone.
As you probably already know, 「何歳ですか」 can be used, but it's not as polite as 「お幾つですか」. Especially in your case, since you are asking about someone's parents, it would be good to show respect by using 「お幾つ」.
Do be careful though, excessive use of 尊敬語 instead of the normal 敬語 can make it seem like you are trying to put more distance between you and the person you are asking about in your relationship with them. If you are pretty close to the person you are asking the age of in both age and relationship, 「何歳」 could be a better option.