Question: If I have two [Parallel Lives](https://gatherer.wizards.com/Pages/Card/Details.aspx?name=Parallel%20Lives), and I have a [Doomed Traveler](https://gatherer.wizards.com/Pages/Card/Details.aspx?name=Doomed%20Traveler) that just died, how many Spirit tokens would I put into play?

Answer: 
Four.

For each instance of Parallel Lives (or Doubling Season) in play, double each instance of a token that would enter play.

Parallel Lives' effect is a replacement effect. It replaces "Put X creature tokens into play under your control" with "Put 2X creature tokens into play under your control."

Thus, two Parallel Lives' would quadruple tokens, three would be eight times, and four would be sixteen. Effectively, for each token placement, replace every token with two tokens for each instance of Parallel Lives.

Appropriate comp rules:

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> 614.5. A replacement effect doesn't invoke itself repeatedly; it gets only one opportunity to affect an event or any modified events that may replace it.
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> Example: A player controls two permanents, each with an ability that reads "If a creature you control would deal damage to a creature or player, it deals double that damage to that creature or player instead." A creature that normally deals 2 damage will deal 8 damage -- not just 4, and not an infinite amount.
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EDIT: Relevant Oracle text mentioned by @AlexP:

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> **9/22/2011** - If you control two Parallel Lives, then the number of tokens created is four times the original number. If you control three, then the number of tokens created is eight times the original number, and so on.
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"If an [EFFECT] would put one or more tokens onto the battlefield [INSTEAD]"

This wording seems confusing. The act of doubling is an effect of the enchantment, therefore to the wording, two Parallel Lives on the field would seem to loop as one would trigger the other into infinite creatures:

The initial spell/effect has you put a creature onto the field.

Parallel Lives (A) has you put an additional token onto the field.

Parallel Lives (B) triggers the same time as (A) resulting in four.

* Going by the text, it seems that it shouldn't end there.

Assuming Parallel Lives (A) would react to (B) having placed tokens

Assuming Parallel Lives (B) would react to (A) having placed a token

"Whenever" vs "If" would have made this card infinite with just the two, as that word is what typically is used to denote potential for an infinite cycle to occur.

If you need proof of the concept go buy two mirrors and put them edge to edge at a 90 degree angle and place an object at the center or place them exactly facing each other with an item in the center. This denotes "Whenever an item appears, create a duplicate of that item," but two mirrors would also catch the appearance of the mirrored appearance.

This is to say the wording didn't covey the comprehensive ruling in simple enough vocabulary/structure, so that is what most player would potentially default to in lack of comprehensive rules to refer to -going on the card text alone. It is understandable to mistake it as a continuous engine.