Sally's dog ate her homework.
Richard's mother gave him lunch.
According to many, it's all good grammar. Linguists refer to this issue as possessive antecedent proscription (PAP). I pretty much reject PAP as a proscriptivist construct similar in nature to the split-inifinitive prohibition.
Even if you buy in to PAP, proscriptivists couldn't ever reject the use of a possessive pronoun with a possessive antecedent. So "Sally's dog ate her homework." has always been good, even to the proscriptivists. They don't claim that the sentence declares the dog to be female and an academic whiz. But "Richard's mother gave him lunch." could provoke a tornado of red ink.
Here is an excellent treatment by Erin Brenner. It also has excellent references for further reading.
Copyediting Tip of the Week: Macbeth's mind
The position you describe, while possible and consistent within itself, does not reflect the majority of writing in support of PAP. I have not seen a description (proscription) that failed to allow the possessive pronoun to refer back to a prior possessive.
This is the way Geoffrey Nunberg paraphrases PAP:
The assumption behind the rule is that a pronoun has to be of the same part of speech as its antecedent. Since possessives are adjectives, the reasoning goes, they can't be followed by pronouns, even if the resulting sentence is perfectly clear.
Arnold Zwicky writes (correctly), citing numerous examples:
6. … Violations of the PAP are frequent, even in the work of careful practiced writers (including the authors of manuals that insist on the PAP), and they go unnoticed.
6.1 Barron's test preparation manual (Ehrenhaft (1998)) … It cares about the PAP so much … But once Barron's gets into extended analysis of particular essays …
6.2. Lunsford & Connors (1999:216), in their excellent handbook, … refer to a “convention” for “maintaining clear pronoun references” … But back on p. 29, when they're giving advice,
6.3 Menand (2003) criticizes CMS15 for failing to mention the PAP. But Menand is jam-packed with violations …
I don't think that PAP deserves any more writing except such writing as points out how destitute of merit and ill-begotten PAP is.
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