How would you classify your parents’ child-rearing styles? What factors might have influenced their approach to parenting?

Since I obviously can't address your individual situation, I will discuss parenting styles in general terms that will help you identify how your parents have approached child-rearing. At the end, I will also mention some of the things that can influence parents.
As proposed by Diana Baumrind, a parenting style isn't any particular tactic, but rather the general approach that a parent takes to guiding, socializing, or controlling a child. Psychologists recognize four such styles.
Authoritarian parenting is what we observe when the parent demands blind obedience and seeks to control behavior through punishment or the threat of punishment. This may include the withdrawal of affection.
Permissive parenting emphasizes emotional warmth and a disinclination to enforce limits. The permissive parent is friendly and encouraging, but fails to step in when the child does something selfish, destructive, rude, or harmful.
Authoritative parenting represents a middle ground where a parent enforces certain standards of behavior, but also provides the child with lots of affection and support for autonomy.
The last parenting style, uninvolved parenting, was added after Baumrind's initial formulation. Uninvolved parents—as the terminology suggests—are parents who provide their kids with very little except the bare minimum of food and shelter.
To apply these definitions to your own parents, you first need to ask yourself if your parents held you to behavioral standards. Was it understood that there were expectations and limits?
If you were pretty much allowed to do what you want—even selfish, harmful, or antisocial behaviors—then by these definitions your parents were either permissive or uninvolved. If they were also affectionate and loving, then they were permissive, not uninvolved.
To distinguish between authoritative and authoritarian parenting, you need to ask if your parents attempted to control you through threats and punishment or by reasoning with you. Did they demand unquestioning obedience? Were they unwilling to entertain discussions about the reasons for rules? If so, they were authoritarian. Did they impose heavy-handed disciplinary actions or punish you for transgressions by becoming cold and withdrawn? Again, these are the characteristics of authoritarian parenting.
Although Baumrind and others initially emphasized the importance of affection in distinguishing between authoritarian and authoritative parenting, this has been criticized as culturally-biased. The parenting style definitions were devised in a predominantly middle-class, European/ American context, and researchers like Ruth Chao note that these definitions don't always provide a good fit for parents from other cultures.
For instance, traditional Chinese parenting resembles authoritarian parenting in some ways, but it is notably affectionate. For this reason, I think the best, least controversial way to distinguish authoritarian from authoritative parenting is to focus on reasoning: Authoritative parents explain and discuss the reason for their rules.
What influences a parent's style? As suggested above, it's partly a question of culture: What you grew up with yourself, and what you see as normative around you. Personality traits also play a role. Some people are more temperamentally inclined toward an authoritarian outlook, for instance.
Scholars like Judith Rich Harris have also made the point that children themselves influence the way their parents react. A child who is frequently defiant and aggressive is going to evoke negative reactions, and if a parent wants to rein him or her in and is having difficulty with an authoritative approach, then the parent may start resorting to harsh punishment and drift into a more authoritarian style. Alternatively, a depressed parent may give up and become permissive, or even uninvolved.
References
See links below, and also -
Chao R. 1994. "Beyond parental control; authoritarian parenting style: Understanding Chinese parenting through the cultural notion of training." Child Development 45: 1111-1119.
Harris, JR. The Nurture Assumption: Why Children Turn Out the Way They Do. The Free Press.
http://persweb.wabash.edu/facstaff/hortonr/articles%20for%20class/baumrind.pdf

https://www.parentingscience.com/parenting-styles.html