Hi Elaine ... how early in the day are you talking about ?
There are some long term refurbishment works taking place on the Poniente promenade, which may cause you some problems on the pavements and pedestrianised areas. But early in the morning, there shouldn't be too much pedestrian congestion in Parque De Elche, or on the road leading down to the harbour. The Levante promenade may be okay early in the morning too. But bear in mind that it's not completely pedestrianised, watch out for delivery vehicles, taxis, motorbikes, etc. It's not unusual to see runners actually down on the beaches, at the water's edge. There is also the pedestrianised Parque De L'Aiguera, it's on an incline but you could always run downhill and get a taxi back up !!
David