I adore my beloved pets, and there's nothing I wouldn't do for them, but my GOD you cannot compare losing a dog or cat (or any other pet) to losing a parent.
No-one is dismissing how upsetting it is to lose a pet, but it is not in the same LEAGUE as losing your mother. As a few people have said, the average lifespan of a pet is 10-13 years, and a human is 7-8 times that! And it's devastating to lose your mother (for most people.) And losing her at a young age is extremely traumatic.
I am so sorry you lost your mother @itsnotthesame and my heart goes out to you - especially if you are young. 
I mean, losing a parent at 25 or 55 is hard, but it's a well documented fact that people are not emotionally or mentally equipped to lose a parent under age of 35. Without giving too much away (as it's potentially identifying,) I also know this from the career I am in.
It's quite possible your colleague didn't mean to be horrid or insensitive, but you do need to let her know that although you understand her grief after losing her dog, you are upset that she is comparing the loss to your mother, especially so soon after your mother's death.
And as a few posters have said, the death of a parent (especially if you are young at the time,) stays with a person for life. The death of a dog or cat that they had for 10-13 years upsets the vast majority of people for a short while, (maybe 6 months to a year,) but it does not have the same long term psychological affect as the death of a parent.
(((HUGS))) for you OP. Take care of yourself.