A freedom of information (FOI) request by the party shows there was an average of 43 ambulance turnarounds that took longer than an hour each day this year.
There is a 20-minute target for each ambulance turnaround, with paramedics expected to hand patients over, fill in forms and then clean the vehicle in preparation for other patients.
The total number of turnarounds of more than an hour was 11,730 to the end of September.
The 2018 total was 13,010 - or an average of 36 per day - and the previous year 9,934 cases were recorded, or 27 per day.
The figure for 2016 was 7,215 turnarounds of more than an hour, or an average of 20 per day.
Scottish Conservative health spokesman Miles Briggs said: “Paramedics are among our most dedicated and hard-working public servants.
“But clearly there are more and more obstacles being placed in their way and that’s having an impact on patient safety, too.
“It’s clearly stated that turnaround times should be around 20 minutes but the hospitals they are going to just aren’t set up to make that happen.
“The last thing ambulance workers want is to be hanging around a hospital when they could be out on the roads responding to emergencies.
“Performance in this has slipped on the SNP Government’s watch and it’s up to Nicola Sturgeon to sort this out if she’s serious about improving the NHS.”
A spokesman for the Scottish Government said: “It is important that ambulance crews are able to hand over patients at hospital as quickly as possible in order for them to be able to respond to emergency calls.
“We have invested additional funding to place ambulance liaison officers in hospitals to reduce delays with ambulance turnaround times and help improve patient flow.
“We have invested almost £900 million in the ambulance service in the last four years and have committed to supporting the training of an additional 1,000 paramedics over the course of this Parliament, building on the 17.6 per cent rise in ambulance service staffing over the last decade.”
A spokesman for the Scottish Ambulance Service said: “We’re committed to keeping turnaround times to a minimum and are working in partnership with all Scottish health boards to help reduce delays.
“We’ve also recruited liaison officers in hospitals to help minimise waiting times.”