You (or a class that you choose) need to change
\parskip. The question (which only you can answer) is "what do you consider to be a paragraph?"
If you are designing a document layout which has visible space between paragraphs, you would start by setting
\parskip and then specify space around headings, lists, math displays etc taking account of the paragraph space already added. If on the other hand you just want to alter an existing layout that was not designed with an option for non-zero paragraph space, then you need to adjust whatever lengths need adjusting, that depends on the class, how many environments it defined. the
parskip package tries to make a reasonable guess at doing this for the standard classes such as
article although as it notes in its documentation this is not really as good as designing a class with this option from the start.
To see why it is not such a simple question, consider some examples.
list items are paragraphs so by default they will be affected, if you want them to not have extra space you need to reset their definition to add less space.
Section headings come between paragraphs, so changing parskip adds space (that would be there if the heading were not present) that may or may not make the space around the heading look too much so you may need to adjust spacing around section headings to add less space, and so it goes on.
Note for section headings you probably want to choose some length based on some under specified notion of what looks good. If you have a large
\parskip and that is added to the space already added around display headings then they may look too spaced out, but conversely if you were to automatically suppress all the
\parskip space near a section heading you could find that section headings are more closely spaced than just a normal paragraph break, which would be odd, so really the entire document design as implemented in the class needs to be reconsidered if paragraph spacing is changed.
That said, for most documents most of the time, choosing a common class where this has been considered in advance (Koma classes and memoir have options for this or the NTG
artikel3 class is a variant of article with non zero parskip) or using
parskip package to adjust the settings in the standard classes works in a reasonable way.