It seems counter-intuitive, but you are not trying to format the text itself. The formatting needs to be applied to the text box that the pull quote is in, so the solution is to use Object Styles rather than paragraph styles.
Access the dialogue box via Window > Styles > Object Styles. As you would with Paragraph or Character styles create a new style and double click to edit it.
InDesign does not have options for padding, you need to set the Inset Spacing for the margins inside the text box...
And then set the Text Wrap Offset for the margins outside the text box...
Remember that these will work in combination - so in my example the inset of 2mm and the wrap of 3mm gives a buffer of 5mm between the pull quote and the text surrounding it. In my opinion anything less than 4mm spacing between pull quotes and body text is too little, it will tend to look crowded once printed. But that's also subject to typeface choices, background vs text frame colour, and so on.
You can also define which Paragraph Style to use within this dialogue box, but the options will need to be set in the Paragraph Styles panel. Depending on your workflow you may find it easier to leave this option blank since any Paragraph Style you choose for the text in the frame will function independently of your Object Styles options.
Also - this is where you want to set the paragraph rules for the lines above and below. This will take a little bit of back and forth, since whatever offset you choose will need to take into account the inset and wrap applied to the text box.