In regards to this sentence why is it ある and not いる for 親 and 責任?
親である限りは、子供に対する責任があると思う。
Um... For 親, it is not the verb ある but rather the verb である which means "to be" like です or だ.
The difference is that である can be used to modify nouns while the 2 latter can't.
責任 (responsibility) here is an inanimate object (non living) so we use ある not いる. Only living things can have いる to identify its existence.
The whole sentence means: I think the limit of being a parents is having the responsibility to direct their child to the future. (Maybe, correct me if I'm wrong)
To describe existence as a normal verb, you have to use いる for animate objects and ある for others.
You always have to use である when it describes some state (i.e., as a copula / linking verb).
いる is also used as a subsidiary verb to describe a progressive action and continuation of state. See this.
In, 親である, ある is the right choice because that part is about "who you are", not about "whether a parent exists".
責任がある refers to the existence of responsibility. ある was chosen because 責任 is not a living thing.