Opener Lyth made an unbeaten 65 and shared in an unbroken 63-run stand for the third wicket with captain Andrew Gale (27 not out) as Yorkshire reached lunch on day one on 124 for two after being put in to bat.
Yorkshire are looking to extend their 10-point lead over nearest challengers Sussex at the top of the Division One standings.
Third-placed Middlesex lost Sam Robson cheaply as they reached 16 for one at lunch in their response to Derbyshire's 385 all out on day two at Derby.
Earlier, home captain Wayne Madsen finished unbeaten on 138 as Derbyshire added an extra 106 runs to their overnight 279 for six, with last two Tim Groenewald (49) and Mark Footitt (19) helping their skipper frustrate Middlesex.
In the other Division One match taking place, relegation-threatened Somerset reduced Warwickshire to 55 for three on the second day at Edgbaston.
After bowling Somerset out for 340 late on yesterday, Warwickshire lost Recordo Gordon (13) and captain Varun Chopra (13) to the spin of George Dockrell this morning while Ian Westwood (eight) fell to Peter Trego as the visitors gained the upper hand.
In Division Two, promotion hopefuls Northamptonshire were on course for a daunting first-innings total against Essex after progressing from 343 for five at the start of the day to 481 for six at lunch.
Andrew Hall was still there on 60 not out while Steven Crook struck a quickfire 50 not out after Rob Keogh had fallen for 38, bowled by Graham Napier for Essex's only success of the morning.
Gloucestershire, in third place, enjoyed the better of the first morning against Kent at Canterbury after they reduced their hosts to 72 for three at the interval.
Sam Northeast, Daniel Bell-Drummond and Brendan Nash all fell fairly cheaply after Kent had opted to bat first before Rob Key (35 not out) set about trying to lead a recovery in partnership with Ben Harmison (13no).
On the opening day of the battle between the bottom two sides in Division Two, last-placed Leicestershire were 113 for three at lunch against Glamorgan at Swansea.
Greg Smith (40) and Niall O'Brien (29) put on 67 for the first wicket before Glamorgan struck back.