Your sentence has a problem. You have enclosed his mother's French in a pair of commas, making it an aside—or parenthetically nonessential information.
If we remove the nonessential information, we end up with:
✘ In particular with his father’s Russian aristocratic background was central.
That is ungrammatical.
In your question, you say two things. You first say that the "subject" is his mother's background. But you then say that it's the background of both his mother and father.
However, it can't be both of those things at the same time. Either it's just his mother's background (and his father's is only an aside), or it's both of their backgrounds.
You have to decide which it is and structure the sentence appropriately.
If only his mother's background is essential information, add another comma before aristocratic:
In particular, his mother’s French, with his father’s Russian, aristocratic background was central.
Now, the comma after particular is no longer the first of a parenthetical pair, but it is used after an introductory (and dependent) clause. This changes the parenthetical information to with his father's Russian.
The following results when it is removed:
✔ In particular, his mother’s French aristocratic background was central.
This is grammatical and the singular background is appropriate. (Note that parenthetical information never affects the syntax of any text that surrounds it. If you want to easily determine what syntax to use in the "main" text, just remove the parenthetical information.)
Note, however, that even though the full version of the sentence is now grammatical, it's also a bit awkward.
One way to make it parse more easily is to actually use parentheses instead of commas:
In particular, his mother’s French aristocratic background was central (his father's background was Russian).
Or, the sentence could be rephrased:
Although his father had a Russian aristocratic background, it was his mother's French aristocratic background that was central.
Finally, if you mean to give equal meaning to both of his parents' backgrounds, then you need to rephrase the sentence.
Two possibilities follow:
In particular, his mother’s French aristocratic background and his father’s Russian aristocratic background were central.
In particular, the aristocratic backgrounds of his parents (his mother's, French; his father's, Russian) were central.
