What is a Parallel Word™?

Question

_{This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.}

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If a word conforms to a special rule, I call it a Parallel Word™.
Use the examples below to find the rule.

$$ % \def\Pad{\P{0.0}} \def\Title{\textbf{ Parallel }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Words}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ } \smash{\lower{29px}\bbox[lightblue]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ wheel}&\text{ truck}\\ \hline \text{ sleep}&\text{wake}\\ \hline \text{ fake}&\text{genuine}\\ \hline \text{ river}&\text{mountain}\\ \hline \text{ hotel}&\text{motel}\\ \hline \text{ banjo}&\text{kazooie}\\ \hline \text{ fusion}&\text{ liquid}\\ \hline \text{ melon}&\text{ fruit}\\ \hline \text{ jolly}&\text{ rogers}\\ \hline \text{ marry}&\text{ parry}\\ \hline \text{ harry}&\text{potter}\\ \hline \text{ bran}&\text{stark}\\ \hline \text{ cobra}&\text{viper}\\ \hline \text{ frog}&\text{ toad}\\ \hline \text{ iron}&\text{maiden}\\ \hline \text{ ingot}&\text{gold}\\ \hline \hline \end{array}$$

CSV version:

Parallel Words™,    Not Parallel Words™

wheel,  truck
sleep,  wake
fake,   genuine
river,  mountain
hotel,  motel
banjo,  kazooie
fusion, liquid
melon,  fruit
jolly,  rogers
marry,  parry
harry,  potter
bran,   stark
cobra,  viper
frog,   toad
iron,   maiden
ingot,  gold

What is the special rule these words conform to?

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Hint:

The frog is cold, and the ingot will make a chain.

Answer 1

A Parallel Word appears to be a word which:

can form another word (or famous name) in English when every letter in the word is Caesar-shifted by the same magnitude.

Like so:

COBRA -- rot-3 --> FREUD
BANJO -- rot-4 --> FERNS
FUSION -- rot-6 --> LAYOUT
WHEEL -- rot-7 --> DOLLS
HOTEL -- rot-7 --> OVALS
RIVER -- rot-9 --> ARENA
SLEEP -- rot-9 --> BUNNY
FAKE -- rot-14 --> TOYS
MELON -- rot-16 --> CUBED
JOLLY -- rot-19 --> CHEER
HARRY -- rot-20 --> BULLS
INGOT -- rot-20 --> CHAIN
MARRY -- rot-20 --> GULLS
FROG -- rot-23 --> COLD
IRON -- rot-23 --> FOLK

The name is suggestive of:

the idea that another word can exist 'in parallel', just with the letters shifted by some amount along the alphabet.

This property is not exhibited by the words listed as Not Parallel.