Why are nude scenes allowed in PG-13 movies?

I believe the difference is between sexual and non-sexual contexts. Female breasts and nipples are allowed (briefly) in PG and PG-13 movies if they are in a non-sexual context.

• Kate Winslet's breasts were bare in the PG-13 movie Titanic. (That was a non-sexual scene, though it was sensually charged.)
• There were also bare female breasts and nipples in the PG-rated 1978 movie "Invasion of the Body Snatchers".
• Julia Roberts bare breasts are visible for several seconds, in shadows, during a scene in the PG film "Pretty Woman" (though most viewers don't even notice them).
• In Return of the Jedi, the green dancer in Jabba the Hutt's palace is wearing a see-through mesh shirt and her nipples are also visible (even though they are dark green).

Directly from the MPAA Wiki:

Nudity is restricted to PG and above, although only brief nudity is permitted in a PG rated film. Nudity that is sexually oriented will generally require an R rating. As of 2010, the MPAA has added a descriptor of "male nudity" to films featuring said content.

PG-13 is obviously above PG. I believe side shots are allowed in PG/PG-13, but nipples and frontal nudity are only allowed in R and above.

This question is impossible to answer as the MPAA is a secretive panel and doesn't need to, and rarely does, explain any of their decision making criteria for rating films.

For background on this, I recommend the documentary This Film Is Not Yet Rated

To perhaps appease the one downvoter, I'll try and answer the question as asked:

Why are nudity scenes allowed in PG-13 movies?

Because the MPAA ratings board allows it.

The best answer is PG-13 was instituted in 1984 as a response to some PG films that some parents felt went too far in content; but that did not have quite enough content to earn an R. Thus if you look at PG-13 films from the first few years, many of them have female nudity. It was almost a running joke for a while that PG-13 meant you were going to see some breasts and get an F-word. This died down quite a bit in the 90s as PG-13 became the main rating used by Hollywood to market their films to the widest audience.

For additional background, going back even before PG-13 was instituted, nudity was not all that uncommon in a PG film. (Even G rated films, if you count innocent skinny dipping kids and male rear ends. I will limit the rest of this answer to refer to female frontal nudity.)

Prior to the 1990s, PG did not mean 'family film' as it does now. You have to realize PG was a replacement for the original 'M' rating (mature) and was described as 'Parental guidance suggested – some material may not be suitable for pre-teenagers.' Thus there was a time when PG actually meant a film had content not suitable for younger children and parents should think before sending their kids off to see it. Examples of PG films with nudity are Logan's Run, Billy Jack, The Outlaw Josie Wales, The Omega Man, Airplane, Coma, Clash of the Titans, The Beastmaster, Sheena, Splash, Sixteen Candles. In some PG films, evan rape or sexual violence was implied or depicted.