Parabolic or Hyperbolic? How can astronomers find the difference between a parabolic and a hyperbolic comet ? What are the criteria that helps them distinguish these ? Can a parabolic comet switch over to become a hyperbolic one and the vice-versa ?
 A: The orbits of comets are distinguished, as you note, by their eccentricities or equivalently by their orbital energies or orbital velocities.


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*If the comet is approaching or receding from the Sun at less than escape velocity, it is on an elliptical trajectory and will end up in a regular elliptic orbit, or just crashing into the Sun. Elliptical comets, which make up the majority of known comets, are divided into "short period" (<200 yrs) and "long period" (>200 yrs). The eccentricity of an elliptical comet is on average less than 1, but can have instantaneous values > 1. 

*If the comet is approaching or receding from the Sun at escape velocity, it is on a parabolic trajectory. It is in the process of being either captured by or ejected from the Sun. As @Vince Mulhollon indicates in his answer, an exactly parabolic orbit is a fleeting state.

*If the comet is approaching or receding from the Sun at greater than escape velocity, it is on a hyperbolic trajectory and won't be back. There is some dispute as to whether hyperbolic comets originating outside the solar system have truly been observed, and if we really know the periods of certain extremely long-term elliptical comets. 
As you ask, parabolic comet can "switch over" to being a hyperbolic comet (and vice versa) if something adds or removes energy from the orbit. For example, gravitational interaction with another solar system body could add energy to a parabolic orbit, and eject the comet from the solar system along a hyperbolic trajectory. 


As to how to tell the difference, just take a set of accurate enough position measurements spaced far enough apart, and calculate the orbital elements. Software will help. 
A: "How can astronomers find the difference between a parabolic and a hyperbolic comet ?  What are the criteria that helps them distinguish these ?"
The "binary" differentiation doesn't really matter.  There are actually three options.  Google or wikipedia for the concept of eccentricity especially WRT orbits.
"Can a parabolic comet switch over to become a hyperbolic one and the vice-versa ?"
In orbital mechanics, anything is possible.  Google or wikipedia for "gravity assist" or "gravitational slingshot" or whatever.  Is it likely?  Well you're about a zillion times more likely to randomly increase your eccentricity than to make it exactly equal "1".
