Current distribution for parallel zeners

Low voltage Zener diodes (about 5V) feature true Zener breakdown, i.e. band-to-band tunneling. The larger the temperature, the more likely the tunneling occurs because the larger the carrier energy (and the smaller also the energy gap).

Therefore, there could be a thermal run-away. In fact, unlike normal diodes (or LED diodes) in forward conduction, small mismatches can lead to large current variations as the curve is very steep in breakdown voltage. The Zener carrying more current will likely heat more (Which will decrease the Zener voltage etc.).

High-voltage Zener diodes (e.g. 10V or more), instead, feature avalanche breakdown. The carriers are accelerated by an electric field, and if they reach enough energy, they will produce more electron-hole pairs, due to impact ionization (which are in turn accelerated by the electric field, etc.). However, the larger the temperature, the larger the lattice vibration and thus the smaller the probability that an electron can gain enough energy between impacts, therefore the less likely they can ionize (i.e. produce another electron-hole pair).

In this way, there will be a negative feedback, that will make the diode initially carrying more current to be less conductive (therefore the current will spread more equally).

The datasheets shows in fact different temperature coefficient for different Zener values.

You could look at a datasheet and notice it mentions a temperature coefficient for zener voltage, which answers your question.

In the case of a 5V1 zener tempco can be positive or negative, so anything can happen...

For a 3V zener, tempco is negative, so you'll have current hogging.

For a 12V zener, tempco is positive, so they would tend to share the current.

You didn't state whether you're using zeners for overvoltage (OV) protection or a voltage source.

I'm going to assume you want OV protection because, if you were making a power supply, you would want to minimize the power wasted by running a semiconductor in its active region. For the purpose of OV protection, a better circuit would be to put several resistor-zener circuits in parallel. Then you have control over both the sign of the coefficient and its magnitude. Think about it from a feedback perspective. As the current increases through a resistor, its voltage increases proportionally. In a resistor-zener series circuit, now that the resistor is eating up more of the total voltage, there is less voltage for the zener to eat, therefore its current will be lower. Using a series resistor, you can always force the system into negative feedback* regardless of the sign of the diode's tempco. Each diode should be in series with its own resistor and in parallel with nothing at all. Put as many pairs as necessary in parallel in order to achieve an acceptable risk profile.

*Do not confuse sign of feedback with sign of tempco.