It's a difficult one isn't it? I may be off the mark but has someone been sowing little seeds of discontent into your head? Be careful with that as it can split many a family in two.
Let's look at this logically,

Your parents work full time but also own a kennel and dogs....... they employ/allow you to run the kennels and also run their breeding programmes, they (I presume) rent, own the land and property, pay the insurance, electric, gas, water, food, maintenance, tax etc of the business and they also pay you and others a wage to run things, in effect you are the manageress (although I agree your wages are way under par for a manageress.)
You have a job which will allow you to take your dogs to work with you and as you say, future children also,
Now, if you were just their employee and not their daughter would it cross your mind for one second that you should become a partner in their kennel name/ business........ I doubt it, you would just move on if the wages were too low for the work you do.
Parents 'generally' will pass on their family business to their children or give a share in the profits, but your parents don't sound to even be financially comfortable yet, as you say they are not frivolous.
You need a pay rise hun, and yes a chat about the future in whether you may inherit the business, or become a future partner, but it's not a given as others have said, depends on the family and it is not a
right because you work hard, plenty of employees work hard and management get all the rewards and plenty of parents are not generous.
They may have no idea, or no intention............. you know them better than us.
But go into any pleasant chats over a coffee with the foresight that if they say 'no' will you leave, will you plod on resentfully?
Please remember you are not the owner of anything here, if a parent has no intention of partnership, you need to just put yourself back in the box of employee only and look to
your future and not what your parents have, hopefully leading to you starting your own business and breeding programme, after all you now have the experience., whichever way it goes.
But please, do not fall out with your parents over money and title, life is too short, if they don't value or see your worth, do it yourself on your own.