The Scarlets were comfortably beaten by Ulster in their European Champions Cup clash on Saturday night.
The Irish side earned a bonus point, with tries from Darren Cave, Ruan Pienaar, Tommy Bowe and Rory Best.
Rhys Priestland kicked three penalties for the Scarlets, who had James Davies and Liam Williams sent to the sin bin.
The defeat leaves the Welsh side in a difficult position in Pool 5,ahead of the return fixtures at Parc y Scarlets next weekend.
Edinburgh and Scarlets had to share the points from what was ultimately a more disappointing 20-20 draw for the Welsh team, who dominated the first half at Murrayfield last night.
Scarlets put Edinburgh on the back foot after Harry Robinson scored the opening try in less than one minute. The second followed by Liam Williams in the first half - both converted by Rhys Priestland, who also added a penalty.
In a more balanced second half, Priestland and Heathcote exchanged penalties before replacement Edinburgh centre Phil Burleigh grabbed a try.
It was converted by Heathcote who could not win the game with a penalty from a foot inside the Scarlets' half.
The second half also began badly for Edinburgh.
However, in the 50th minute, Heathcote narrowed the gap with his second penalty.
Scarlets had lost some of their zip by this stage and the game settled largely in midfield as the Edinburgh line out continued to misfire.
The pendulum swung Edinburgh's way in the 72 minute.
With John Barclay sent to the sin bin, a driven line-out saw the ball released to Burleigh, who barged over. Heathcote's conversion tied the scores.
Scarlets Head coach Wayne Pivac is giving fullback Liam Williams until Thursday to prove his fitness for Saturday's Pro12 match against Treviso.
Welsh international Williams missed the 42-12 defeat against Leinster on the weekend after injuring his groin during his sides warm up in Dublin.
Although the Scarlets are confident he'll recover in time to be fit for the visit of the Italians.
Centre Regan King is set to miss a second consecutive game due to injury.
Pivac says he's keen for his Scarlets side to make amends after the heavy defeat to Leinster.
We let the fans down and we let ourselves down and we need to show we are a better side than that this weekend when we play Treviso.
We created scoring opportunities in Dublin, but we didn't make the most of them. Individual errors cost us and, if we have an error rate like that against the top teams, that's what happens.
The Scarlets were denied victory in their first game of the PRO12 season, as Ulster grabbed a late try to draw 32-32 in Llanelli.
First half tries from Harry Robinson, Rory Pitman and Gareth Davies put the Scarlets comfortably ahead at the break, and they extended their lead to 27-15 with another Rhys Priestland penalty.
Scrum half Gareth Davies got his second try to earn the Scarlets a bonus point, but the Llanelli side were down to 14 men in the closing minutes after Liam Williams was shown a second yellow card - and Ulster made them pay.
Andrew Trimble crossed the line with just a minute to play and Ian Humphreys' conversion levelled the scores at the Parc y Scarlets.
Bristol have signed centre Gareth Maule from the Scarlets in time for the new Championship season.
Wales Under 20 cap Maule played over 100 times for the Scarlets and also had stints at Newport Gwent Dragons, Ebbw Vale, Newport RFC and Cross Keys.
I depart with some invaluable memories, gathered in my years with this special club and feel honoured to have worn the famous Scarlet jersey on so many occasions and to join the list of centurions is something that makes me immensely proud. Being able to be part of so many memorable occasions and huge fixtures was a huge privilege.
A special mention must go to the supporters. From the day I arrived I felt welcome and was quick to realise that the Scarlets has its very own style and the fans are no different!
The Scarlets centre has been out of action since February but had hoped to make a return to action before the end of the season.Read the full story ›
The Scarlets have withstood a late Connacht fightback to get a bonus point win in the Pro12 this afternoon. Tries from Jonathan Davies and Josh Turnbull, and two touchdowns from Gareth Davies and helped seal the victory.
Jonathan Davies's try gave the Scarlets the lead at home against Connacht two minutes into the game. A conversion and two penalties from Rhys Priestland leaves the Scarlets 13-6 ahead at half time.
The Wales fly-half has committed to the club, ending speculation he was set to become the latest Wales international to leave the country.Read the full story ›
Wales fly-half Rhys Priestland has extended his contract with the Scarlets, putting an end to recent speculation over his rugby future.
The 26-year-old had been linked with clubs in England and France, as well as the possibility of a contract with the Welsh Rugby Union.