Is it illegal to burn yard trash (leaves, grass, wood, etc) in your yard?

yes--in city limits. I'll look up the law in a sec.

ETA: Sec. 5:96. - Burning trash, weeds, fields, etc. (Parish)

Any person, other than authorized personnel of the department of public works acting under authority of section 5:31(b), starting or lighting any fires to burn trash (other than in a proper container which will prevent the spread of any ashes therefrom), weeds or fields, or to clear land within the parish and outside the corporate limits of any municipality, shall be guilty of a misdemeanor.

(Parish Code 1962, Title 5, § 83)

Get a fire pit or barrel