Word for country in which parents were born

Solution 1:

Perhaps motherland

one's native land or, sometimes, the land of one's ancestors; a country thought of as originator or source

Similarly fatherland

a person's native land or country; the land or country of one's ancestors

Solution 2:

It's worth looking at some figures from Google Books here...

1 - predominantly OUR
our/my motherland 79,600 / 22,200
our/my fatherland 105,000 / 53,800
our/my homeland 160,000 / 88,500

2 - predominantly MY
our/my birthland 105 / 431
our/my native land 130,000 / 234,000
land of our/my birth 130,000 / 256,000

The above results suggest to me that mother/father/homeland are primarily associated with identifying shared biological and cultural origins. When an individual simply wants to identify his personal background, he's likely to avoid terms which are primarily nationalistic/militaristic...

defend the/our/my motherland 8,470 / 8,060 / 186
fight for the/our/my fatherland 48,500 / 1,320 / 237
the/our/my homeland security 1,780,000 / 122,000 / 600

Quite apart from those connotations (which are presumably unwanted in OP's context), there's a literal/figurative clash in forms such as my father's motherland, my parent's fatherland, etc. Plus there's obviously no reason to mention your parents' origins at all unless they're different to yours. But if you have to, the general preference is quite clear...

my parents' motherland 5
my parents' fatherland 4
my parents' homeland 1,150 <--- (as per Mari-Lou's earlier comment)
my parents' birthland 2
my parents' native land 158
land of my parents' birth 78