Safety of DSA key parameters sharing I'm looking for a solution to use in a context where I need to be able to generate new asymmetric key pairs quickly (using a widely recognized algorithm, and EC-DSA is not applicable). It sounds like DSA would be the adequate solution.

According to the documentation, just generating DSA parameters once allows then to generate a large number of separate keys pairs only requiring the step of selecting a random private key 'x' for each of them. FIPS 186-3 says : 


  the intended signatory shall first obtain appropriate domain
  parameters, either by generating the domain parameters itself, or by
  obtaining domain parameters that another entity has generated


Is there any security issue I'm missing before reusing the same domain parameters for a large number of keys ? How frequently ought the domain parameters be changed ? (There's some reference around to the possibility for a CA to use the same domain parameters for all the certificates it issues)
In DSA (and ECDSA), it is possible and common to share the same domain parameters, across multiples users. AFAIK (and according to common wisdom, including FIPS recommendations) this introduce no known security weakness. The only common reasons to change domain parameters are to increase key size, or purposely introduce on interoperability barrier (e.g. to force an upgrade).

One advantage of fixed domain parameters $(p,q,g)$ in DSA is that it simplifies the choice of a user's key: basically one selects a random private $x\in[1, q–1]$ and computes the public $y=g^x\bmod p$.

As pointed in this other answer, sharing domain parameters increase odds that several users share the same key. However, for a good RNG, $2^n$ users and the lowest security level in FIPS 186, odds that it happens are less then $2^{160-2⋅n}$; that's negligible even for one key per day per human on our planet during a century. And if the RNG is flawed, all bets are off anyway.