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Since I'm not entirely content with the answers to date, here's my take - everyone seems to agree on the basics of forces generated by pressure differentials. If you took a rigid uncooked spaghetto with a cross section of $A$, the case is quite clear - on the cylinder's base in your mouth, a force of $p_{in} A$ is trying to push the spaghetto out, and on the other end, a force of $p_{out} A$ pushes it inwards. The forces generated by the pressures on the sides even out.

It becomes more confusing when referring to cooked spaghetti since a) the cooked spaghetto won't stick straight out from your mouth and b) we intuitively don't want it to transmit forces because "you can't push on a string".

However, these complications do not change the underlying principle. Imagine cutting the spaghetto right outside your mouth while you keep sucking on the inner part. Clearly, the free spaghetto wouldn't feel any net force from air pressure after the cut and just float away/fall down. By cutting, you have changed the total picture by adding $p_{out} A$ orthogonal to the cut surface, but removed the push of $p_{in} A$, so before there must have been a net force on this surface before if there is none afterwards.

I know that it's not the qusestion, but in this case it is much simpler to look at the basic thermodynamics - the system is basically a microcanonical ensemble, with $E = p_{out} V_{out} + p_{in} V_{in}$ under the constraint that $V_{in} + V_{out} = \mathrm{const}$, and if $p_{in} < p_{out}$, the minimum energy (and thus equilibrium) state is clearly that with $V_{out}=0$.