Dear Daniel,

Your term "Likert scale data" is a bit ambiguous. Some people use it to refer to individual LIkert response questions whilst others use it to refer to a collection of such questions (often about the same psychological construct). If it is the latter then there is the secondary issue of whether it has been validated as a scale (e.g. using Cronbach's alpha reliability analysis). If it has been validated then you could either add the individual question scores together to create a scale variable or create a regression scale variable using the components scores obtained from the first eigenvalue in a principal component analysis.

Once you have scale variables you can explore your hypotheses, e.g. starting with scatter plots then deciding what kind of correlation or regression analysis to attempt.

If you cannot create scale variables you may need to enter the complex world of structural equation modelling.

Building upon the previous answer...It sounds like the scales you are using are established in your discipline (e.g. valid and reliable). Assuming that, the question then becomes how you intend on creating the variables you are talking about.

Are you aggregating specific subscales to determine parenting styles? What are the other variables you mentioned? Are they also aggregated from scales? 

There have been so many questions here about creating scales from LIkert-score items that I have compiled a set of resources on this topic:

https://www.researchgate.net/post/How_can_one_help_people_create_simple_scales_from_Likert-scored_items