You can visit the parent as many times as you want! The first 3 times, you get a Souveniour from the parent, but after that you can keep visiting - only you don't get any more souveniors.
Well, you can only get a present 3 times, but you can visit as many times as you like!!! Ifr you visit again, she will say something like: "I'm sorry dear, I haven't got any more Tama Treats to give you. Maybe you should visit your grandparents instead."
1st time: Bicycle (though unforuntaly, you cannot use it, just view it in your souvonier list)
2nd time: Pencil (Though this time, you can use it!!! yay!)