There is a large difference between tone deafness and an undeveloped voice. Unless your parents are musicians, comments like that can be hurtful and can stifle musical exploration and creativity.
Tone deafness is actually quite serious and is as it suggests - an inability to distinguish between certain sounds. This is akin to color-blindness, where a person may be unable to see a red object if it is resting in some green grass.
Tone deafness has nothing to do with the timbre or quality of your voice. If you are musically untrained, you would be expected to have an aspirate (breathy) sound with a range of about an octave total. Just because you may not sound "good" initially has no relevance as to how your voice would sound if you were given a little information, some encouragement, and some practice.
In a strictly technical sense, Perfect Pitch is the ability to match a given pitch, not the ability to reproduce any pitch without external suggestion as so many ascribe. The latter definition more correctly applies to the term "Absolute Pitch", which has not yet gained universal acceptance. If you are able to differentiate between two notes and can tell which is higher / lower, then you are not tone deaf.
Relative pitch is being able to identify and reproduce a given note within the context of another given note or chord. If you were to think about it in terms of color, you would say: "I know this color is blue because that color is red.
People with perfect pitch would be able to give you the color blue if you gave them the color blue.
Alternatively, people with "absolute pitch" do not need to compare colors to know what they are, nor to do they need to be dictated. They would be able to create the color blue with no external reference. This is akin to the way most people experience color - they just identify it. Relative, perfect, and absolute pitch do not need to be mysteries as so many believe, and I maintain that the last of those three can be taught.
But, I must now digress as this subject can quickly lead into tangents and unwarranted discussion.
Having a "terrible singing voice" stems from what I referenced earlier: being untrained in creating efficient vocal production. Learning some simple techniques and vocal development exercises will help you not only use your air more efficiently, but begin developing fine coordination of your vocal folds.
Many famous musicians - from composers and performers to conductors, all have fantastic ears but horrible singing voices. Listen to Bernstein sing through The Rite of Spring, or Keith Jarrett sing...all the time. Terrible. Both fantastic musicians and were clearly not tone deaf.
BOTTOM LINE: Tone deafness and voice timbre are completely unrelated. Mechanics of vocal production can be easily taught. Get out there and start singin'!