Well, I had to look up 'andirons' myself: according to Wikipedia, these are metal supports (aka 'fire-dogs'), used to provide an airspace under logs while they burn. The front ends may be quite ornately molded, like this set from Tiffany's.
I guess 'covert' may be in the sense of 'covert feathers', which cover (streamline, and insulate) the base of the flight-feathers on birds' wings; although it could also be in the sense of a camouflaged shelter used for e.g. reconnaissance, hunting, or bird-watching. So your idea that the (front base of the) andirons might have moldings shaped like asparagus(-spears?), isn't a bad one.
A remote possibility is that the "asparagus tops" (if spears) are poking out of a pot which is sitting on the hearth, in front of the fire, to boil them for dinner. Objections to that, are that cooking would be done in the kitchen, rather than the parlour (it's a big house, yes? It would surely have a separate kitchen), and that winter is anyway not at all the right season for eating asparagus (unless it had spent the last 6 months in a preserving-jar)!
So I think @Virote_Considon is right, that a pot of asparagus ferns (which develop in late summer/autumn, if you don't harvest the spears in June-ish) has been set on the hearth, in front of the andirons, purely as decoration. Having a room(s) in your house that you use only rarely (and fill with ornaments), is a sign of wealth after all, so the 'best parlour' (and its fireplace) might well not be in daily — or even weekly — use. And the narrator is just peeking round the door, he isn't being invited into that room by his host...
Probably is so. My issue is with:
a) asparagus-shaped ends (which "tops" could refer to) are a usual shape for quality fireplace objects like those mentioned, eg the tongs:

and also it would help explain why they glistened (due to the multi-faceted top of the object, reflecting far more light).
Still, i cannot include "covert" in that sense :/ So yes, it likely is what you and Virote said, about asparagus ferns being obstructing the view (but imo it is a strangely written sentence, unless such a sight would be quite common; can't know

In the end i suppose he wants to show that even mostly hindered by the asparagus leaves, those objects still shine considerably.
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