I'm after words which begin with "para", where this is not a prefix. One of those brain tangents I sometimes take...

One each, to make it fair. Go!

parallax?

(Any word where the "par-" part is the prefix, rather than para-, will meet your conditions.)

(I was going to put 'parable', but no.)

paraffle, Scot., an ostentatious display :-)

In that case, excluding "par" prefixes (I want to type prefices) as well.

parading away from the obscure, how about parade?
[fr. F. parer, to adorn]

It meets the criteria. Fwiw, I started my own meanderings with paragon.

bzzzz

AHD (OED concurs) has perhaps from Greek parakonan, to sharpen : para-, alongside; see para–1 + akon, whetstone
(cf. It. paragonare)

Paradise?

Indeed. I did suspect many of the para words would have some deep dark derivation from para meaning alongside. And that the discussion would head that way. I guess I don't see the para in paragon as a prefix because

the definition is not simply "alongside a/the gon"
it was attached so long ago that it can not be unaffixed
the average person wouldn't see it a prefix
I was really just trying to avoid words like paralegal and paramedic
it's my thread and I'm making up the rules :p

I'm not making the rules (hi doc), but if you go back far enough
(see AHD) you'll find 'around the wall'.
<shrugs>

Paraffin
The par- is a prefix but it's not from the Greek. Well, maybe not really a prefix.

Parador

Parament

Paramo

Parang

Parasang

There are more if you include words with par- followed by a vowel, e.g., parous.

I just grabbed my brick and mortar AHD and looked. I know, so last millennium.

paradox?

paranja

(simplistically) para

re: paradox

my bad.

Paradiddle! Thanks to eta for this (from the past--I still have it bookmarked).

That's also true of paradise.

Parakeet (???)

parallel :Latin paralllus, from Greek paralllos : para-, beside; see para–1 + allln, of one another (from allos, other; see al-1 in Appendix I). Yes.
paralyze

parallel: beside ... other Cool! Thanks.